5 \chapter{Minimum Spanning Trees}
9 The problem of finding a minimum spanning tree of a weighted graph is one of the
10 best studied problems in the area of combinatorial optimization since its birth.
11 Its colorful history (see \cite{graham:msthistory} and \cite{nesetril:history} for the full account)
12 begins in~1926 with the pioneering work of Bor\o{u}vka
13 \cite{boruvka:ojistem}\foot{See \cite{nesetril:boruvka} for an English translation with commentary.},
14 who studied primarily an Euclidean version of the problem related to planning
15 of electrical transmission lines (see \cite{boruvka:networks}), but gave an efficient
16 algorithm for the general version of the problem. As it was well before the dawn of graph
17 theory, the language of his paper was complicated, so we will better state the problem
18 in contemporary terminology:
20 \proclaim{Problem}Given an undirected graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$,
21 find its minimum spanning tree, defined as follows:
24 For a given graph~$G$ with weights $w:E(G)\rightarrow {\bb R}$:
26 \:A~subgraph $H\subseteq G$ is called a \df{spanning subgraph} if $V(H)=V(G)$.
27 \:A~\df{spanning tree} of~$G$ is any spanning subgraph of~$G$ that is a tree.
28 \:For any subgraph $H\subseteq G$ we define its \df{weight} $w(H):=\sum_{e\in E(H)} w(e)$.
29 When comparing two weights, we will use the terms \df{lighter} and \df{heavier} in the
31 \:A~\df{minimum spanning tree (MST)} of~$G$ is a spanning tree~$T$ such that its weight $w(T)$
32 is the smallest possible among all the spanning trees of~$G$.
33 \:For a disconnected graph, a \df{(minimum) spanning forest (MSF)} is defined as
34 a union of (minimum) spanning trees of its connected components.
37 Bor\o{u}vka's work was further extended by Jarn\'\i{}k \cite{jarnik:ojistem}, again in
38 mostly geometric setting. He has discovered another efficient algorithm. However, when
39 computer science and graph theory started forming in the 1950's and the
40 spanning tree problem was one of the central topics of the flourishing new
41 disciplines, the previous work was not well known and the algorithms had to be
42 rediscovered several times.
44 In the next 50 years, several significantly faster algorithms were discovered, ranging
45 from the $\O(m\timesbeta(m,n))$ time algorithm by Fredman and Tarjan \cite{ft:fibonacci},
46 over algorithms with inverse-Ackermann type complexity by Chazelle \cite{chazelle:ackermann}
47 and Pettie \cite{pettie:ackermann}, to an~algorithm by Pettie \cite{pettie:optimal}
48 whose time complexity is provably optimal. Frequently, the most important ingredients
49 were advances in data structures used to represent the graph.
51 In the upcoming chapters, we will explore this colorful universe of MST algorithms.
52 We will meet the canonical works of the classics, the clever ideas of their successors,
53 various approaches to the problem including randomization and solving of important
54 special cases. At several places, we will try to contribute our little stones to this
57 %--------------------------------------------------------------------------------
59 \section{Basic properties}\id{mstbasics}%
61 In this section, we will examine the basic properties of spanning trees and prove
62 several important theorems which will serve as a~foundation for our MST algorithms.
63 We will mostly follow the theory developed by Tarjan in~\cite{tarjan:dsna}.
65 For the whole section, we will fix a~connected graph~$G$ with edge weights~$w$ and all
66 other graphs will be spanning subgraphs of~$G$. We will use the same notation
67 for the subgraphs as for the corresponding sets of edges.
69 First of all, let us show that the weights on edges are not necessary for the
70 definition of the MST. We can formulate an equivalent characterization using
71 an~ordering of edges instead.
73 \defnn{Heavy and light edges}\id{heavy}%
74 Let~$T$ be a~spanning tree. Then:
76 \:For vertices $x$ and $y$, let $T[x,y]$ denote the (unique) path in~$T$ joining $x$ with~$y$.
77 \:For an edge $e=xy$ we will call $T[e]:=T[x,y]$ the \df{path covered by~$e$} and
78 the edges of this path \df{edges covered by~$e$}.
79 \:An edge~$e$ is called \df{light with respect to~$T$} (or just \df{$T$-light}) if it covers a~heavier edge, i.e., if there
80 is an~edge $f\in T[e]$ such that $w(f) > w(e)$.
81 \:An edge~$e$ is called \df{$T$-heavy} if it covers a~lighter edge.
85 Edges of the tree~$T$ cover only themselves and thus they are neither heavy nor light.
86 The same can happen if an~edge outside~$T$ covers only edges of the same weight,
87 but this will be rare because all edge weights will be usually distinct.
89 \lemman{Light edges}\id{lightlemma}%
90 Let $T$ be a spanning tree. If there exists a $T$-light edge, then~$T$
94 If there is a $T$-light edge~$e$, then there exists an edge $e'\in T[e]$ such
95 that $w(e')>w(e)$. Now $T-e'$ ($T$~with the edge~$e'$ removed) is a forest of two trees with endpoints of~$e$
96 located in different components, so adding $e$ to this forest must restore
97 connectivity and $T':=T-e'+e$ is another spanning tree with weight $w(T')
98 = w(T)-w(e')+w(e) < w(T)$. Hence $T$ could not have been minimum.
101 \figure{mst2.eps}{278pt}{An edge exchange as in the proof of Lemma~\ref{lightlemma}}
103 The converse of this lemma is also true and to prove it, we will once again use
104 the technique of transforming trees by \df{exchanges of edges.} In the proof of the
105 lemma, we have made use of the fact that whenever we exchange an edge~$e$ of
106 a~spanning tree for another edge~$f$ covered by~$e$, the result is again
107 a~spanning tree. In fact, it is possible to transform any spanning tree
108 to any other spanning tree by a sequence of exchanges.
110 \lemman{Exchange property for trees}\id{xchglemma}%
111 Let $T$ and $T'$ be spanning trees of a common graph. Then there exists
112 a sequence of edge exchanges that transforms $T$ to~$T'$. More formally,
113 there exists a sequence of spanning trees $T=T_0,T_1,\ldots,T_k=T'$ such that
114 $T_{i+1}=T_i - e_i + e_i^\prime$ where $e_i\in T_i$ and $e_i^\prime\in T'$.
117 By induction on $d(T,T'):=\vert T\symdiff T'\vert$. When $d(T,T')=0$,
118 both trees are identical and no exchanges are needed. Otherwise, the trees are different,
119 but as they have the same number of edges, there must exist an edge $e'\in T'\setminus T$.
120 The cycle $T[e']+e'$ cannot be wholly contained in~$T'$, so there also must
121 exist an edge $e\in T[e']\setminus T'$. Exchanging $e$ for~$e'$ yields a spanning
122 tree $T^*:=T-e+e'$ such that $d(T^*,T')=d(T,T')-2$. Now we can apply the induction
123 hypothesis to $T^*$ and $T'$ to get the rest of the exchange sequence.
126 \figure{mst1.eps}{295pt}{One step of the proof of Lemma~\ref{xchglemma}}
128 \>In some cases, a~much stronger statement is true:
130 \lemman{Monotone exchanges}\id{monoxchg}%
131 Let $T$ be a spanning tree such that there are no $T$-light edges and $T'$
132 be an arbitrary spanning tree. Then there exists a sequence of edge exchanges
133 transforming $T$ to~$T'$ such that the weight of the tree does not decrease in any step.
136 We improve the argument from the previous proof, refining the induction step.
137 When we exchange $e\in T$ for $e'\in T'\setminus T$ such that $e\in T[e']$,
138 the weight never drops, since $e'$ is not a $T$-light edge and therefore
139 $w(e') \ge w(e)$, so $w(T^*)=w(T)-w(e)+w(e')\ge w(T)$.
141 To keep the induction going, we have to make sure that there are still no light
142 edges with respect to~$T^*$. In fact, it is enough to avoid such edges in
143 $T'\setminus T^*$, since these are the only edges considered by the induction
144 steps. To accomplish that, we replace the so far arbitrary choice of $e'\in T'\setminus T$
145 by picking the lightest such edge.
147 Let us consider an edge $f\in T'\setminus T^*$. We want to show that $f$ is not
148 $T^*$-light, i.e., that it is heavier than all edges on $T^*[f]$. The path $T^*[f]$ is
149 either identical to the original path $T[f]$ (if $e\not\in T[f]$) or to $T[f] \symdiff C$,
150 where $C$ is the cycle $T[e']+e'$. The former case is trivial, in the latter we have
151 $w(f)\ge w(e')$ due to the choice of~$e'$ and all other edges on~$C$ are lighter
152 than~$e'$ as $e'$ was not $T$-light.
155 This lemma immediately implies that Lemma \ref{lightlemma} works in both directions:
157 \thmn{Minimality of spanning trees}\id{mstthm}%
158 A~spanning tree~$T$ is minimum iff there is no $T$-light edge.
161 If~$T$ is minimum, then by Lemma~\ref{lightlemma} there are no $T$-light
163 Conversely, when $T$ is a spanning tree without $T$-light edges
164 and $T_{min}$ is an arbitrary minimum spanning tree, then according to the Monotone
165 exchange lemma (\ref{monoxchg}) there exists a non-decreasing sequence
166 of exchanges transforming $T$ to $T_{min}$, so $w(T)\le w(T_{min})$
167 and thus $T$~is also minimum.
170 In general, a single graph can have many minimum spanning trees (for example
171 a complete graph on~$n$ vertices with unit edge weights has $n^{n-2}$
172 minimum spanning trees according to the Cayley's formula \cite{cayley:trees}).
173 However, as the following theorem shows, this is possible only if the weight
174 function is not injective.
176 \thmn{Uniqueness of MST}%
177 If all edge weights are distinct, then the minimum spanning tree is unique.
180 Consider two minimum spanning trees $T_1$ and~$T_2$. According to the previous
181 theorem, there are no light edges with respect to neither of them, so by the
182 Monotone exchange lemma (\ref{monoxchg}) there exists a sequence of non-decreasing
183 edge exchanges going from $T_1$ to $T_2$. As all edge weights all distinct,
184 these edge exchanges must be in fact strictly increasing. On the other hand,
185 we know that $w(T_1)=w(T_2)$, so the exchange sequence must be empty and indeed
186 $T_1$ and $T_2$ must be identical.
187 \looseness=1 %%HACK%%
191 When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
192 its unique minimum spanning tree.
194 Also the following trivial lemma will be often invaluable:
196 \lemman{Edge removal}
197 Let~$G$ be a~graph with distinct edge weights and $e \in G\setminus\mst(G)$.
198 Then $\mst(G-e) = \mst(G)$.
201 The tree $T=\mst(G)$ is also a~MST of~$G-e$, because every $T$-light
202 edge in~$G-e$ is also $T$-light in~$G$. Then we apply the uniqueness of
206 \paran{Comparison oracles}\id{edgeoracle}%
207 To simplify the description of MST algorithms, we will assume that the weights
208 of all edges are distinct and that instead of numeric weights we are given a~comparison oracle.
209 The oracle is a~function that answers questions of type ``Is $w(e)<w(f)$?'' in
210 constant time. This will conveniently shield us from problems with representation
211 of real numbers in algorithms and in the few cases where we need a more concrete
212 input, we will explicitly state so.
214 In case the weights are not distinct, we can easily break ties by comparing some
215 unique identifiers of edges. According to our characterization of minimum spanning
216 trees, the unique MST of the new graph will still be a~MST of the original graph.
217 Sometimes, we could be interested in finding all solutions, but as this is an~uncommon
218 problem, we will postpone it until Section \ref{kbestsect}. For the time being,
219 we will always assume distinct weights.
222 If all edge weights are distinct and $T$~is an~arbitrary spanning tree, then every edge of~$G$
223 is either $T$-heavy, or $T$-light, or contained in~$T$.
225 \paran{Monotone isomorphism}%
226 Another useful consequence of the Minimality theorem is that whenever two graphs are isomorphic and the
227 isomorphism preserves the relative order of weights, the isomorphism applies to their MST's as well:
230 A~\df{monotone isomorphism} between two weighted graphs $G_1=(V_1,E_1,w_1)$ and
231 $G_2=(V_2,E_2,w_2)$ is a bijection $\pi: V_1\rightarrow V_2$ such that
232 for each $u,v\in V_1: uv\in E_1 \Leftrightarrow \pi(u)\pi(v)\in E_2$ and
233 for each $e,f\in E_1: w_1(e)<w_1(f) \Leftrightarrow w_2(\pi[e]) < w_2(\pi[f])$.
235 \lemman{MST of isomorphic graphs}\id{mstiso}%
236 Let~$G_1$ and $G_2$ be two weighted graphs with distinct edge weights and $\pi$
237 a~monotone isomorphism between them. Then $\mst(G_2) = \pi[\mst(G_1)]$.
240 The isomorphism~$\pi$ maps spanning trees to spanning trees bijectively and it preserves
241 the relation of covering. Since it is monotone, it preserves the property of
242 being a light edge (an~edge $e\in E(G_1)$ is $T$-light $\Leftrightarrow$
243 the edge $\pi[e]\in E(G_2)$ is~$f[T]$-light). Therefore by the Minimality Theorem
244 (\ref{mstthm}), $T$ is the MST of~$G_1$ if and only if $\pi[T]$ is the MST of~$G_2$.
247 %--------------------------------------------------------------------------------
249 \section{The Red-Blue meta-algorithm}
251 Most MST algorithms can be described as special cases of the following procedure
252 (again following Tarjan \cite{tarjan:dsna}):
254 \algn{Red-Blue Meta-Algorithm}\id{rbma}%
256 \algin A~graph $G$ with an edge comparison oracle (see \ref{edgeoracle})
257 \:At the beginning, all edges are colored black.
258 \:Apply rules as long as possible:
259 \::Either pick a cut~$C$ such that its lightest edge is not blue \hfil\break and color this edge blue, \cmt{Blue rule}
260 \::or pick a cycle~$C$ such that its heaviest edge is not red \hfil\break and color this edge \rack{blue.}{red.\hfil} \cmt{Red rule}
261 \algout Minimum spanning tree of~$G$ consisting of edges colored blue.
265 This procedure is not a proper algorithm, since it does not specify how to choose
266 the rule to apply. We will however prove that no matter how the rules are applied,
267 the procedure always stops and it gives the correct result. Also, it will turn out
268 that each of the classical MST algorithms can be described as a specific way
269 of choosing the rules in this procedure, which justifies the name meta-algorithm.
272 We will denote the unique minimum spanning tree of the input graph by~$T_{min}$.
273 We intend to prove that this is also the output of the procedure.
276 Let us prove that the meta-algorithm is correct. First we show that the edges colored
277 blue in any step of the procedure always belong to~$T_{min}$ and that the edges colored
278 red are guaranteed to be outside~$T_{min}$. Then we demonstrate that the procedure
279 always stops. Some parts of the proof will turn out to be useful in the upcoming chapters,
280 so we will state them in a~slightly more general way.
282 \lemman{Blue lemma, also known as the Cut rule}\id{bluelemma}%
283 The lightest edge of every cut is contained in the MST.
286 By contradiction. Let $e$ be the lightest edge of a cut~$C$.
287 If $e\not\in T_{min}$, then there must exist an edge $e'\in T_{min}$ that is
288 contained in~$C$ (take any pair of vertices separated by~$C$: the path
289 in~$T_{min}$ joining these vertices must cross~$C$ at least once). Exchanging
290 $e$ for $e'$ in $T_{min}$ yields an even lighter spanning tree since
294 \lemman{Red lemma, also known as the Cycle rule}\id{redlemma}%
295 An~edge~$e$ is not contained in the MST iff it is the heaviest on some cycle.
298 The implication from the left to the right follows directly from the Minimality
299 theorem: if~$e\not\in T_{min}$, then $e$~is $T_{min}$-heavy and so it is the heaviest
300 edge on the cycle $T_{min}[e]+e$.
302 We will prove the other implication again by contradiction. Suppose that $e$ is the heaviest edge of
303 a cycle~$C$ and that $e\in T_{min}$. Removing $e$ causes $T_{min}$ to split
304 to two components, let us call them $T_x$ and~$T_y$. Some vertices of~$C$ now lie in~$T_x$, the
305 others in~$T_y$, so there must exist in edge $e'\ne e$ such that its endpoints lie in different
306 components. Since $w(e')<w(e)$, exchanging $e$ for~$e'$ yields a~spanning tree lighter than
310 \figure{mst-rb.eps}{289pt}{Proof of the Blue (left) and Red (right) lemma}
312 \lemman{Black lemma}%
313 As long as there exists a black edge, at least one rule can be applied.
316 Assume that $e=xy$ is a black edge. Let us define~$M$ as the set of vertices
317 reachable from~$x$ using only blue edges. If $y$~lies in~$M$, then $e$ together
318 with some blue path between $x$ and $y$ forms a cycle and $e$~must be the heaviest
319 edge on this cycle. This holds because all blue edges have been already proven
320 to be in $T_{min}$ and there can be no $T_{min}$-light edges.
321 In this case, we can apply the Red rule.
323 On the other hand, if $y\not\in M$, then the cut formed by all edges between $M$
324 and $V\setminus M$ contains no blue edges, therefore we can use the Blue rule.
327 \figure{mst-bez.eps}{295pt}{Configurations in the proof of the Black lemma}
330 We will use $\delta(M)$ to denote the cut separating~$M$ from its complement.
331 That is, $\delta(M) = E \cap (M \times (V\setminus M))$. We will also abbreviate
332 $\delta(\{v\})$ as~$\delta(v)$.
334 \thmn{Red-Blue correctness}%
335 For any selection of rules, the Red-Blue procedure stops and the blue edges form
336 the minimum spanning tree of the input graph.
339 To prove that the procedure stops, let us notice that no edge is ever recolored,
340 so we must run out of black edges after at most~$m$ steps. Recoloring
341 to the same color is avoided by the conditions built in the rules, recoloring to
342 a different color would mean that the edge would be both inside and outside~$T_{min}$
343 due to our Red and Blue lemmata.
345 When no further rules can be applied, the Black lemma guarantees that all edges
346 are colored, so by the Blue lemma all blue edges are in~$T_{min}$ and by the Red
347 lemma all other (red) edges are outside~$T_{min}$. Thus the blue edges are exactly~$T_{min}$.
351 The MST problem is a~special case of the problem of finding the minimum basis
352 of a~weighted matroid. Surprisingly, when we modify the Red-Blue procedure to
353 use the standard definitions of cycles and cuts in matroids, it will always
354 find the minimum basis. Some of the other MST algorithms also easily generalize to
355 matroids and in some sense matroids are exactly the objects where ``the greedy approach works''. We
356 will however not pursue this direction in our work, referring the reader to the Oxley's monograph
357 \cite{oxley:matroids} instead.
359 %--------------------------------------------------------------------------------
361 \section{Classical algorithms}\id{classalg}%
363 The three classical MST algorithms (Bor\o{u}vka's, Jarn\'\i{}k's, and Kruskal's) can be easily
364 stated in terms of the Red-Blue meta-algorithm. For each of them, we first show the general version
365 of the algorithm, then we prove that it gives the correct result and finally we discuss the time
366 complexity of various implementations.
368 \paran{Bor\o{u}vka's algorithm}%
369 The oldest MST algorithm is based on a~simple idea: grow a~forest in a~sequence of
370 iterations until it becomes connected. We start with a~forest of isolated
371 vertices. In each iteration, we let each tree of the forest select the lightest
372 edge of those having exactly one endpoint in the tree (we will call such edges
373 the \df{neighboring edges} of the tree). We add all such edges to the forest and
374 proceed with the next iteration.
376 \algn{Bor\o{u}vka \cite{boruvka:ojistem}, Choquet \cite{choquet:mst}, Florek et al.~\cite{florek:liaison}, Sollin \cite{sollin:mst}}
378 \algin A~graph~$G$ with an edge comparison oracle.
379 \:$T\=$ a forest consisting of vertices of~$G$ and no edges.
380 \:While $T$ is not connected:
381 \::For each component $T_i$ of~$T$, choose the lightest edge $e_i$ from the cut
382 separating $T_i$ from the rest of~$T$.
383 \::Add all $e_i$'s to~$T$.
384 \algout Minimum spanning tree~$T$.
387 \lemma\id{boruvkadrop}%
388 In each iteration of the algorithm, the number of trees in~$T$ decreases by at least
392 Each tree gets merged with at least one of its neighbors, so each of the new trees
393 contains two or more original trees.
397 The algorithm stops in $\O(\log n)$ iterations.
400 The Bor\o{u}vka's algorithm outputs the MST of the input graph.
403 In every iteration of the algorithm, $T$ is a blue subgraph,
404 because every addition of some edge~$e_i$ to~$T$ is a straightforward
405 application of the Blue rule. We stop when the blue subgraph is connected, so
406 we do not need the Red rule to explicitly exclude edges.
408 It remains to show that adding the edges simultaneously does not
409 produce a~cycle. Consider the first iteration of the algorithm where $T$ contains a~cycle~$C$. Without
410 loss of generality we can assume that:
411 $$C=T_1[u_1,v_1]\,v_1u_2\,T_2[u_2,v_2]\,v_2u_3\,T_3[u_3,v_3]\, \ldots \,T_k[u_k,v_k]\,v_ku_1.$$
412 Each component $T_i$ has chosen its lightest incident edge~$e_i$ as either the edge $v_iu_{i+1}$
413 or $v_{i-1}u_i$ (indexing cyclically). Suppose that $e_1=v_1u_2$ (otherwise we reverse the orientation
414 of the cycle). Then $e_2=v_2u_3$ and $w(e_2)<w(e_1)$ and we can continue in the same way,
415 getting $w(e_1)>w(e_2)>\ldots>w(e_k)>w(e_1)$, which is a~contradiction.
416 (Note that distinctness of edge weights was crucial here.)
419 \lemma\id{boruvkaiter}%
420 Each iteration can be carried out in time $\O(m)$.
423 We assign a label to each tree and we keep a mapping from vertices to the
424 labels of the trees they belong to. We scan all edges, map their endpoints
425 to the particular trees and for each tree we maintain the lightest incident edge
426 so far encountered. Instead of merging the trees one by one (which would be too
427 slow), we build an auxiliary graph whose vertices are the labels of the original
428 trees and edges correspond to the chosen lightest inter-tree edges. We find the connected
429 components of this graph, and these determine how are the original labels translated
434 The Bor\o{u}vka's algorithm finds the MST in time $\O(m\log n)$.
437 Follows from the previous lemmata.
440 \paran{Jarn\'\i{}k's algorithm}%
441 The next algorithm, discovered independently by Jarn\'\i{}k, Prim and Dijkstra, is similar
442 to the Bor\o{u}vka's algorithm, but instead of the whole forest it concentrates on
443 a~single tree. It starts with a~single vertex and it repeatedly extends the tree
444 by the lightest neighboring edge until the tree spans the whole graph.
446 \algn{Jarn\'\i{}k \cite{jarnik:ojistem}, Prim \cite{prim:mst}, Dijkstra \cite{dijkstra:mst}}\id{jarnik}%
448 \algin A~graph~$G$ with an edge comparison oracle.
449 \:$T\=$ a single-vertex tree containing an~arbitrary vertex of~$G$.
450 \:While there are vertices outside $T$:
451 \::Pick the lightest edge $uv$ such that $u\in V(T)$ and $v\not\in V(T)$.
453 \algout Minimum spanning tree~$T$.
457 The Jarn\'\i{}k's algorithm computes the MST of the input graph.
460 If~$G$ is connected, the algorithm always stops. In every step of
461 the algorithm, $T$ is always a blue tree. because Step~4 corresponds to applying
462 the Blue rule to the cut $\delta(T)$ separating~$T$ from the rest of the given graph. We need not care about
463 the remaining edges, since for a connected graph the algorithm always stops with the right
464 number of blue edges.
468 The most important part of the algorithm is finding the \em{neighboring edges.}
469 In a~straightforward implementation, searching for the lightest neighboring
470 edge takes $\Theta(m)$ time, so the whole algorithm runs in time $\Theta(mn)$.
472 We can do much better by using a binary
473 heap to hold all neighboring edges. In each iteration, we find and delete the
474 minimum edge from the heap and once we expand the tree, we insert the newly discovered
475 neighboring edges to the heap and delete the neighboring edges that became
476 internal to the new tree. Since there are always at most~$m$ edges in the heap,
477 each heap operation takes $\O(\log m)=\O(\log n)$ time. For every edge, we perform
478 at most one insertion and at most one deletion, so we spend $\O(m\log n)$ time in total.
479 From this, we can conclude:
482 The Jarn\'\i{}k's algorithm computes the MST of a~given graph in time $\O(m\log n)$.
485 We will show several faster implementations in Section \ref{iteralg}.
487 \paran{Kruskal's algorithm}%
488 The last of the three classical algorithms processes the edges of the
489 graph~$G$ greedily. It starts with an~empty forest and it takes the edges of~$G$
490 in order of their increasing weights. For every edge, it checks whether its
491 addition to the forest produces a~cycle and if it does not, the edge is added.
492 Otherwise, the edge is dropped and not considered again.
494 \algn{Kruskal \cite{kruskal:mst}}
496 \algin A~graph~$G$ with an edge comparison oracle.
497 \:Sort edges of~$G$ by their increasing weights.
498 \:$T\=\hbox{an empty spanning subgraph}$.
499 \:For all edges $e$ in their sorted order:
500 \::If $T+e$ is acyclic, add~$e$ to~$T$.
501 \::Otherwise drop~$e$.
502 \algout Minimum spanning tree~$T$.
506 The Kruskal's algorithm returns the MST of the input graph.
509 In every step, $T$ is a forest of blue trees. Adding~$e$ to~$T$
510 in step~4 applies the Blue rule on the cut separating some pair of components of~$T$ ($e$ is the lightest,
511 because all other edges of the cut have not been considered yet). Dropping~$e$ in step~5 corresponds
512 to the Red rule on the cycle found ($e$~must be the heaviest, since all other edges of the
513 cycle have been already processed). At the end of the algorithm, all edges are colored,
514 so~$T$ must be the~MST.
518 Except for the initial sorting, which in general requires $\Theta(m\log m)$ time, the only
519 other non-trivial operation is the detection of cycles. What we need is a~data structure
520 for maintaining connected components, which supports queries and edge insertion.
521 This is closely related to the well-known Disjoint Set Union problem:
523 \problemn{Disjoint Set Union, DSU}
524 Maintain an~equivalence relation on a~finite set under a~sequence of operations \<Union>
525 and \<Find>. The \<Find> operation tests whether two elements are equivalent and \<Union>
526 joins two different equivalence classes into one.
529 We can maintain the connected components of our forest~$T$ as equivalence classes. When we want
530 to add an~edge~$uv$, we first call $\<Find>(u,v)$ to check if both endpoints of the edge lie in
531 the same component. If they do not, addition of this edge connects both components into one,
532 so we perform $\<Union>(u,v)$ to merge the equivalence classes.
534 Tarjan \cite{tarjan:setunion} has shown that there is a~data structure for the DSU problem
535 of surprising efficiency:
537 \thmn{Disjoint Set Union, Tarjan \cite{tarjan:setunion}}\id{dfu}%
538 Starting with a~trivial equivalence with single-element classes, a~sequence of operations
539 comprising of $n$~\<Union>s intermixed with $m\ge n$~\<Find>s can be processed in time
540 $\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is a~certain inverse of the Ackermann's function
541 (see Definition \ref{ackerinv}).
544 Using this data structure, we get the following bound:
547 The Kruskal's algorithm finds the MST of a given graph in time $\O(m\log n)$.
548 If the edges are already sorted by their weights, the time drops to
549 $\O(m\timesalpha(m,n))$.
552 We spend $\O(m\log n)$ time on sorting, $\O(m\timesalpha(m,n))$ on processing the sequence
553 of \<Union>s and \<Find>s, and $\O(m)$ on all other work.
557 The cost of the \<Union> and \<Find> operations is of course dwarfed by the complexity
558 of sorting, so a much simpler (at least in terms of its analysis) data
559 structure would be sufficient, as long as it has $\O(\log n)$ amortized complexity
560 per operation. For example, we can label vertices with identifiers of the
561 corresponding components and always relabel the smaller of the two components.
563 We will study dynamic maintenance of connected components in more detail in Chapter~\ref{dynchap}.
565 %--------------------------------------------------------------------------------
567 \section{Contractive algorithms}\id{contalg}%
569 While the classical algorithms are based on growing suitable trees, they
570 can be also reformulated in terms of edge contraction. Instead of keeping
571 a~forest of trees, we can keep each tree contracted to a single vertex.
572 This replaces the relatively complex tree-edge incidencies by simple
573 vertex-edge incidencies, potentially speeding up the calculation at the
574 expense of having to perform the contractions.
576 We will show a contractive version of the Bor\o{u}vka's algorithm
577 in which these costs are carefully balanced, leading for example to
578 a linear-time algorithm for MST in planar graphs.
580 There are two definitions of edge contraction that differ when an edge of
581 a~triangle is contracted. Either we unify the other two edges to a single edge
582 or we keep them as two parallel edges, leaving us with a~multigraph. We will
583 use the multigraph version and we will show that we can easily reduce the multigraph
584 to a~simple graph later. (See \ref{contract} for the exact definitions.)
586 We only need to be able to map edges of the contracted graph to the original
587 edges, so we let each edge carry a unique label $\ell(e)$ that will be preserved by
590 \lemman{Flattening a multigraph}\id{flattening}%
591 Let $G$ be a multigraph and $G'$ its subgraph obtaining by removing loops
592 and replacing each bundle of parallel edges by its lightest edge.
593 Then $G'$~has the same MST as~$G$.
596 Every spanning tree of~$G'$ is a spanning tree of~$G$. In the other direction:
597 Loops can be never contained in a spanning tree. If there is a spanning tree~$T$
598 containing a~removed edge~$e$ parallel to an edge~$e'\in G'$, exchanging $e'$
599 for~$e$ makes~$T$ lighter. (This is indeed the multigraph version of the Red
600 lemma applied to a~two-edge cycle, as we will see in \ref{multimst}.)
603 \algn{Contractive version of Bor\o{u}vka's algorithm}\id{contbor}
605 \algin A~graph~$G$ with an edge comparison oracle.
607 \:$\ell(e)\=e$ for all edges~$e$. \cmt{Initialize the labels.}
609 \::For each vertex $v_k$ of~$G$, let $e_k$ be the lightest edge incident to~$v_k$.
610 \::$T\=T\cup \{ \ell(e_1),\ldots,\ell(e_n) \}$.\hfil\break\cmt{Remember labels of all selected edges.}
611 \::Contract all edges $e_k$, inheriting labels and weights.\foot{In other words, we will ask the comparison oracle for the edge $\ell(e)$ instead of~$e$.}
612 \::Flatten $G$ (remove parallel edges and loops).
613 \algout Minimum spanning tree~$T$.
617 For the analysis of the algorithm, we will denote the graph considered by the algorithm
618 at the beginning of the $i$-th iteration by $G_i$ (starting with $G_0=G$) and the number
619 of vertices and edges of this graph by $n_i$ and $m_i$ respectively. A~single iteration
620 of the algorithm will be called a~\df{Bor\o{u}vka step}.
623 The $i$-th Bor\o{u}vka step can be carried out in time~$\O(m_i)$.
626 The only non-trivial parts are steps 6 and~7. Contractions can be handled similarly
627 to the unions in the original Bor\o{u}vka's algorithm (see \ref{boruvkaiter}):
628 We build an~auxiliary graph containing only the selected edges~$e_k$, find
629 connected components of this graph and renumber vertices in each component to
630 the identifier of the component. This takes $\O(m_i)$ time.
632 Flattening is performed by first removing the loops and then bucket-sorting the edges
633 (as ordered pairs of vertex identifiers) lexicographically, which brings parallel
634 edges together. The bucket sort uses two passes with $n_i$~buckets, so it takes
635 $\O(n_i+m_i)=\O(m_i)$.
639 The Contractive Bor\o{u}vka's algorithm finds the MST of the input graph in
640 time $\O(\min(n^2,m\log n))$.
643 As in the original Bor\o{u}vka's algorithm, the number of iterations is $\O(\log n)$.
644 When combined with the previous lemma, it gives an~$\O(m\log n)$ upper bound.
646 To get the $\O(n^2)$ bound, we observe that the number of trees in the non-contracting
647 version of the algorithm drops at least by a factor of two in each iteration (Lemma \ref{boruvkadrop})
648 and the same must hold for the number of vertices in the contracting version.
649 Therefore $n_i\le n/2^i$. While the number of edges need not decrease geometrically,
650 we still have $m_i\le n_i^2$ as the graphs~$G_i$ are simple (we explicitly removed multiple
651 edges and loops at the end of the previous iteration). Hence the total time spent
652 in all iterations is $\O(\sum_i n_i^2) = \O(\sum_i n^2/4^i) = \O(n^2)$.
655 On planar graphs, the algorithm runs much faster:
657 \thmn{Contractive Bor\o{u}vka on planar graphs}\id{planarbor}%
658 When the input graph is planar, the Contractive Bor\o{u}vka's algorithm runs in
662 Let us refine the previous proof. We already know that $n_i \le n/2^i$. We will
663 prove that when~$G$ is planar, the $m_i$'s are decreasing geometrically. We know that every
664 $G_i$ is planar, because the class of planar graphs is closed under edge deletion and
665 contraction. Moreover, $G_i$~is also simple, so we can use the standard bound on
666 the number of edges of planar simple graphs (see for example \cite{diestel:gt}) to get $m_i\le 3n_i \le 3n/2^i$.
667 The total time complexity of the algorithm is therefore $\O(\sum_i m_i)=\O(\sum_i n/2^i)=\O(n)$.
671 There are several other possibilities how to find the MST of a planar graph in linear time.
672 For example, Matsui \cite{matsui:planar} has described an algorithm based on simultaneously
673 working on the graph and its topological dual. The advantage of our approach is that we do not need
674 to construct the planar embedding explicitly. We will show another simpler linear-time algorithm
675 in section~\ref{minorclosed}.
678 To achieve the linear time complexity, the algorithm needs a very careful implementation,
679 but we defer the technical details to section~\ref{bucketsort}.
681 \paran{General contractions}%
682 Graph contractions are indeed a~very powerful tool and they can be used in other MST
683 algorithms as well. The following lemma shows the gist:
685 \lemman{Contraction of MST edges}\id{contlemma}%
686 Let $G$ be a weighted graph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
687 produced by contracting~$e$ in~$G$, and $\pi$ the bijection between edges of~$G-e$ and
688 their counterparts in~$G/e$. Then $\mst(G) = \pi^{-1}[\mst(G/e)] + e.$
691 % We seem not to need this lemma for multigraphs...
692 %If there are any loops or parallel edges in~$G$, we can flatten the graph. According to the
693 %Flattening lemma (\ref{flattening}), the MST stays the same and if we remove a parallel edge
694 %or loop~$f$, then $\pi(f)$ would be removed when flattening~$G/e$, so $f$ never participates
696 The right-hand side of the equality is a spanning tree of~$G$. Let us denote it by~$T$ and
697 the MST of $G/e$ by~$T'$. If $T$ were not minimum, there would exist a $T$-light edge~$f$ in~$G$
698 (by the Minimality Theorem, \ref{mstthm}). If the path $T[f]$ covered by~$f$ does not contain~$e$,
699 then $\pi[T[f]]$ is a path covered by~$\pi(f)$ in~$T'$. Otherwise $\pi(T[f]-e)$ is such a path.
700 In both cases, $f$ is $T'$-light, which contradicts the minimality of~$T'$. (We do not have
701 a~multigraph version of the theorem, but the direction we need is a~straightforward edge exchange,
702 which obviously works in multigraphs as well as in simple graphs.)
706 In the Contractive Bor\o{u}vka's algorithm, the role of the mapping~$\pi^{-1}$ is of course played by the edge labels~$\ell$.
708 \paran{A~lower bound}%
709 Finally, we will show a family of graphs for which the $\O(m\log n)$ bound on time complexity
710 is tight. The graphs do not have unique weights, but they are constructed in a way that
711 the algorithm never compares two edges with the same weight. Therefore, when two such
712 graphs are monotonically isomorphic (see~\ref{mstiso}), the algorithm processes them in the same way.
715 A~\df{distractor of order~$k$,} denoted by~$D_k$, is a path on $n=2^k$~vertices $v_1,\ldots,v_n$,
716 where each edge $v_iv_{i+1}$ has its weight equal to the number of trailing zeroes in the binary
717 representation of the number~$i$. The vertex $v_1$ is called a~\df{base} of the distractor.
719 \figure{distractor.eps}{\epsfxsize}{A~distractor $D_3$ and its evolution (bold edges are contracted)}
722 Alternatively, we can use a recursive definition: $D_0$ is a single vertex, $D_{k+1}$ consists
723 of two disjoint copies of~$D_k$ joined by an edge of weight~$k$.
726 A~single iteration of the contractive algorithm reduces the distractor~$D_k$ to a~graph isomorphic with~$D_{k-1}$.
729 Each vertex~$v$ of~$D_k$ is incident with a single edge of weight~1. The algorithm therefore
730 selects all weight~1 edges and contracts them. This produces a~graph that is
731 equal to $D_{k-1}$ with all weights increased by~1, which does not change the relative order of edges.
735 A~\df{hedgehog}~$H_{a,k}$ is a graph consisting of $a$~distractors $D_k^1,\ldots,D_k^a$ of order~$k$
736 together with edges of a complete graph on the bases of these distractors. The additional edges
737 have arbitrary weights that are heavier than the edges of all the distractors.
739 \figure{hedgehog.eps}{\epsfxsize}{A~hedgehog $H_{5,2}$ (quills bent to fit in the picture)}
742 A~single iteration of the contractive algorithm reduces~$H_{a,k}$ to a~graph isomorphic with $H_{a,k-1}$.
745 Each vertex is incident with an edge of some distractor, so the algorithm does not select
746 any edge of the complete graph. Contraction therefore reduces each distractor to a smaller
747 distractor (modulo an additive factor in weight) and it leaves the complete graph intact.
748 The resulting graph is monotonely isomorphic to $H_{a,k-1}$.
751 When we set the parameters appropriately, we get the following lower bound:
753 \thmn{Lower bound for Contractive Bor\o{u}vka}%
754 For each $n$ there exists a graph on $\Theta(n)$ vertices and $\Theta(n)$ edges
755 such that the Contractive Bor\o{u}vka's algorithm spends time $\Omega(n\log n)$ on it.
758 Consider the hedgehog $H_{a,k}$ for $a=\lceil\sqrt n\rceil$ and $k=\lceil\log_2 a\rceil$.
759 It has $a\cdot 2^k = \Theta(n)$ vertices and ${a \choose 2} + a\cdot 2^k = \Theta(a^2) + \Theta(a^2) = \Theta(n)$ edges
762 By the previous lemma, the algorithm proceeds through a sequence of hedgehogs $H_{a,k},
763 H_{a,k-1}, \ldots, H_{a,0}$ (up to monotone isomorphism), so it needs a logarithmic number of iterations plus some more
764 to finish on the remaining complete graph. Each iteration runs on a graph with $\Omega(n)$
765 edges as every $H_{a,k}$ contains a complete graph on~$a$ vertices.
768 %--------------------------------------------------------------------------------
770 \section{Lifting restrictions}
772 In order to have a~simple and neat theory, we have introduced several restrictions
773 on the graphs in which we search for the MST. As in some rare cases we are going to
774 meet graphs that do not fit into this simplified world, let us quickly examine what
775 happens when the restrictions are lifted.
777 \paran{Disconnected graphs}\id{disconn}%
778 The basic properties of minimum spanning trees and the algorithms presented in
779 this chapter apply to minimum spanning forests of disconnected graphs, too.
780 The proofs of our theorems and the steps of our algorithms are based on adjacency
781 of vertices and existence of paths, so they are always local to a~single
782 connected component. The Bor\o{u}vka's and Kruskal's algorithm need no changes,
783 the Jarn\'\i{}k's algorithm has to be invoked separately for each component.
785 We can also extend the notion of light and heavy edges with respect
786 to a~tree to forests: When an~edge~$e$ connects two vertices lying in the same
787 tree~$T$ of a~forest~$F$, it is $F$-heavy iff it is $T$-heavy (similarly
788 for $F$-light). Edges connecting two different trees are always considered
789 $F$-light. Again, a~spanning forest~$F$ is minimum iff there are no $F$-light
792 \paran{Multigraphs}\id{multimst}%
793 All theorems and algorithms from this chapter work for multigraphs as well,
794 only the notation sometimes gets crabbed, which we preferred to avoid. The Minimality
795 theorem and the Blue rule stay unchanged. The Red rule is naturally extended to
796 self-loops (which are never in the MST) and two-edge cycles (where the heavier
797 edge can be dropped) as already suggested in the Flattening lemma (\ref{flattening}).
799 \paran{Multiple edges of the same weight}\id{multiweight}%
800 In case when the edge weights are not distinct, the characterization of minimum
801 spanning trees using light edges is still correct, but the MST is no longer unique
802 (as already mentioned, there can be as much as~$n^{n-2}$ MST's).
804 In the Red-Blue procedure, we have to avoid being too zealous. The Blue lemma cannot
805 guarantee that when a~cut contains multiple edges of the minimum weight, all of them
806 are in the MST. It will however tell that if we pick one of these edges, an~arbitrary
807 MST can be modified to another MST that contains this edge. Therefore the Blue rule
808 will change to ``Pick a~cut~$C$ such that it does not contain any blue edge and color
809 one of its lightest edges blue.'' The Red lemma and the Red rule can be handled
810 in a~similar manner. The modified algorithm will be then guaranteed to find one of
813 The Kruskal's and Jarn\'\i{}k's algorithms keep working. This is however not the case of the
814 Bor\o{u}vka's algorithm, whose proof of correctness in Lemma \ref{borcorr} explicitly referred to
815 distinct weights and indeed, if they are not distinct, the algorithm will occasionally produce
816 cycles. To avoid the cycles, the ties in edge weight comparisons have to be broken in a~systematic
817 way. The same applies to the contractive version of this algorithm.