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 of 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, giving 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 another algorithm by Pettie \cite{pettie:optimal}
48 whose time complexity is provably optimal.
50 In the upcoming chapters, we will explore this colorful universe of MST algorithms.
51 We will meet the canonical works of the classics, the clever ideas of their successors,
52 various approaches to the problem including randomization and solving of important
53 special cases. At several places, we will try to contribute our little stones to this
56 %--------------------------------------------------------------------------------
58 \section{Basic properties}\id{mstbasics}%
60 In this section, we will examine the basic properties of spanning trees and prove
61 several important theorems which will serve as a~foundation for our MST algorithms.
62 We will mostly follow the theory developed by Tarjan in~\cite{tarjan:dsna}.
64 For the whole section, we will fix a~connected graph~$G$ with edge weights~$w$ and all
65 other graphs will be spanning subgraphs of~$G$. We will use the same notation
66 for the subgraphs as for the corresponding sets of edges.
68 First of all, let us show that the weights on edges are not necessary for the
69 definition of the MST. We can formulate an equivalent characterization using
70 an ordering of edges instead.
72 \defnn{Heavy and light edges}\id{heavy}%
73 Let~$T$ be a~spanning tree. Then:
75 \:For vertices $x$ and $y$, let $T[x,y]$ denote the (unique) path in~$T$ joining $x$ and~$y$.
76 \:For an edge $e=xy$ we will call $T[e]:=T[x,y]$ the \df{path covered by~$e$} and
77 the edges of this path \df{edges covered by~$e$}.
78 \: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
79 is an edge $f\in T[e]$ such that $w(f) > w(e)$.
80 \:An edge~$e$ is called \df{$T$-heavy} if it covers a~lighter edge.
84 Edges of the tree~$T$ cover only themselves and thus they are neither heavy nor light.
85 The same can happen if an~edge outside~$T$ covers only edges of the same weight,
86 but this will be rare because all edge weights will be usually distinct.
88 \lemman{Light edges}\id{lightlemma}%
89 Let $T$ be a spanning tree. If there exists a $T$-light edge, then~$T$
93 If there is a $T$-light edge~$e$, then there exists an edge $e'\in T[e]$ such
94 that $w(e')>w(e)$. Now $T-e'$ is a forest of two trees with endpoints of~$e$
95 located in different components, so adding $e$ to this forest must restore
96 connectivity and $T':=T-e'+e$ is another spanning tree with weight $w(T')
97 = w(T)-w(e')+w(e) < w(T)$. Hence $T$ could not have been minimum.
100 \figure{mst2.eps}{278pt}{An edge exchange as in the proof of Lemma~\ref{lightlemma}}
102 The converse of this lemma is also true and to prove it, we will once again use
103 technique of transforming trees by \df{exchanges} of edges. In the proof of the
104 lemma, we have made use of the fact that whenever we exchange an edge~$e$ of
105 a spanning tree for another edge~$f$ covered by~$e$, the result is again
106 a spanning tree. In fact, it is possible to transform any spanning tree
107 to any other spanning tree by a sequence of exchanges.
109 \lemman{Exchange property for trees}\id{xchglemma}%
110 Let $T$ and $T'$ be spanning trees of a common graph. Then there exists
111 a sequence of edge exchanges that transforms $T$ to~$T'$. More formally,
112 there exists a sequence of spanning trees $T=T_0,T_1,\ldots,T_k=T'$ such that
113 $T_{i+1}=T_i - e_i + e_i^\prime$ where $e_i\in T_i$ and $e_i^\prime\in T'$.
116 By induction on $d(T,T'):=\vert T\symdiff T'\vert$. When $d(T,T')=0$,
117 both trees are identical and no exchanges are needed. Otherwise, the trees are different,
118 but as they are of the same size, there must exist an edge $e'\in T'\setminus T$.
119 The cycle $T[e']+e'$ cannot be wholly contained in~$T'$, so there also must
120 exist an edge $e\in T[e']\setminus T'$. Exchanging $e$ for~$e'$ yields a spanning
121 tree $T^*:=T-e+e'$ such that $d(T^*,T')=d(T,T')-2$ and we can apply the induction
122 hypothesis to $T^*$ and $T'$ to get the rest of the exchange sequence.
125 \figure{mst1.eps}{295pt}{One step of the proof of Lemma~\ref{xchglemma}}
127 \lemman{Monotone exchanges}\id{monoxchg}%
128 Let $T$ be a spanning tree such that there are no $T$-light edges and $T'$
129 be an arbitrary spanning tree. Then there exists a sequence of edge exchanges
130 transforming $T$ to~$T'$ such that the weight does not decrease in any step.
133 We improve the argument from the previous proof, refining the induction step.
134 When we exchange $e\in T$ for $e'\in T'\setminus T$ such that $e\in T[e']$,
135 the weight never drops, since $e'$ is not a $T$-light edge and therefore
136 $w(e') \ge w(e)$, so $w(T^*)=w(T)-w(e)+w(e')\ge w(T)$.
138 To keep the induction going, we have to make sure that there are still no light
139 edges with respect to~$T^*$. In fact, it is enough to avoid such edges in
140 $T'\setminus T^*$, since these are the only edges considered by the induction
141 steps. To accomplish that, we replace the so far arbitrary choice of $e'\in T'\setminus T$
142 by picking the lightest such edge.
144 Now consider an edge $f\in T'\setminus T^*$. We want to show that $f$ is not
145 $T^*$-light, i.e., that it is heavier than all edges on $T^*[f]$. The path $T^*[f]$ is
146 either equal to the original path $T[f]$ (if $e\not\in T[f]$) or to $T[f] \symdiff C$,
147 where $C$ is the cycle $T[e']+e'$. The former case is trivial, in the latter one
148 $w(f)\ge w(e')$ due to the choice of $e'$ and all other edges on~$C$ are lighter
149 than~$e'$ as $e'$ was not $T$-light.
152 \thmn{Minimality of spanning trees}\id{mstthm}%
153 A~spanning tree~$T$ is minimum iff there is no $T$-light edge.
156 If~$T$ is minimum, then by Lemma~\ref{lightlemma} there are no $T$-light
158 Conversely, when $T$ is a spanning tree without $T$-light edges
159 and $T_{min}$ is an arbitrary minimum spanning tree, then according to the Monotone
160 exchange lemma (\ref{monoxchg}) there exists a non-decreasing sequence
161 of exchanges transforming $T$ to $T_{min}$, so $w(T)\le w(T_{min})$
162 and thus $T$~is also minimum.
165 In general, a single graph can have many minimum spanning trees (for example
166 a complete graph on~$n$ vertices and unit edge weights has $n^{n-2}$
167 minimum spanning trees according to the Cayley's formula \cite{cayley:trees}).
168 However, as the following theorem shows, this is possible only if the weight
169 function is not injective.
171 \thmn{Uniqueness of MST}%
172 If all edge weights are distinct, then the minimum spanning tree is unique.
175 Consider two minimum spanning trees $T_1$ and~$T_2$. According to the previous
176 theorem, there are no light edges with respect to neither of them, so by the
177 Monotone exchange lemma (\ref{monoxchg}) there exists a sequence of non-decreasing
178 edge exchanges going from $T_1$ to $T_2$. As all edge weights all distinct,
179 these edge exchanges must be in fact strictly increasing. On the other hand,
180 we know that $w(T_1)=w(T_2)$, so the exchange sequence must be empty and indeed
181 $T_1$ and $T_2$ must be identical.
185 When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
186 its unique minimum spanning tree.
188 \paran{Comparison oracles}\id{edgeoracle}%
189 To simplify the description of MST algorithms, we will assume that the weights
190 of all edges are distinct and that instead of numeric weights we are given a~comparison oracle.
191 The oracle is a~function that answers questions of type ``Is $w(e)<w(f)$?'' in
192 constant time. This will conveniently shield us from problems with representation
193 of real numbers in algorithms and in the few cases where we need a more concrete
194 input, we will explicitly state so.
196 In case the weights are not distinct, we can easily break ties by comparing some
197 unique identifiers of edges. According to our characterization of minimum spanning
198 trees, the unique MST of the new graph will still be a~MST of the original graph.
201 If all edge weights are distinct and $T$~is an~arbitrary tree, then for every tree~$T$ all edges are
202 either $T$-heavy, or $T$-light, or contained in~$T$.
204 \paran{Monotone isomorphism}%
205 Another useful consequence is that whenever two graphs are isomorphic and the
206 isomorphism preserves the relative order of weights, the isomorphism applies to their MST's as well:
209 A~\df{monotone isomorphism} of two weighted graphs $G_1=(V_1,E_1,w_1)$ and
210 $G_2=(V_2,E_2,w_2)$ is a bijection $\pi: V_1\rightarrow V_2$ such that
211 for each $u,v\in V_1: uv\in E_1 \Leftrightarrow \pi(u)\pi(v)\in E_2$ and
212 for each $e,f\in E_1: w_1(e)<w_1(f) \Leftrightarrow w_2(\pi[e]) < w_2(\pi[f])$.
214 \lemman{MST of isomorphic graphs}\id{mstiso}%
215 Let~$G_1$ and $G_2$ be two weighted graphs with distinct edge weights and $\pi$
216 their monotone isomorphism. Then $\mst(G_2) = \pi[\mst(G_1)]$.
219 The isomorphism~$\pi$ maps spanning trees onto spanning trees and it preserves
220 the relation of covering. Since it is monotone, it preserves the property of
221 being a light edge (an~edge $e\in E(G_1)$ is $T$-light $\Leftrightarrow$
222 the edge $\pi[e]\in E(G_2)$ is~$f[T]$-light). Therefore by the Minimality Theorem
223 (\ref{mstthm}), $T$ is the MST of~$G_1$ if and only if $\pi[T]$ is the MST of~$G_2$.
226 %--------------------------------------------------------------------------------
228 \section{The Red-Blue meta-algorithm}
230 Most MST algorithms can be described as special cases of the following procedure
231 (again following Tarjan \cite{tarjan:dsna}):
233 \algn{Red-Blue Meta-Algorithm}\id{rbma}%
235 \algin A~graph $G$ with an edge comparison oracle (see \ref{edgeoracle})
236 \:In the beginning, all edges are colored black.
237 \:Apply rules as long as possible:
238 \::Either pick a cut~$C$ such that its lightest edge is not blue \hfil\break and color this edge blue, \cmt{Blue rule}
239 \::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}
240 \algout Minimum spanning tree of~$G$ consisting of edges colored blue.
244 This procedure is not a proper algorithm, since it does not specify how to choose
245 the rule to apply. We will however prove that no matter how the rules are applied,
246 the procedure always stops and gives the correct result. Also, it will turn out
247 that each of the classical MST algorithms can be described as a specific way
248 of choosing the rules in this procedure, which justifies the name meta-algorithm.
251 We will denote the unique minimum spanning tree of the input graph by~$T_{min}$.
252 We intend to prove that this is also the output of the procedure.
255 Let us prove that the meta-algorithm is correct. First we show that the edges colored
256 blue in any step of the procedure always belong to~$T_{min}$ and that edges colored
257 red are guaranteed to be outside~$T_{min}$. Then we demonstrate that the procedure
258 always stops. We will prefer a~slightly more general formulation of the lemmata, which will turn out
259 to be useful in the future chapters.
261 \lemman{Blue lemma, also known as the Cut rule}\id{bluelemma}%
262 The lightest edge of every cut is contained in the MST.
265 By contradiction. Let $e$ be the lightest edge of a cut~$C$.
266 If $e\not\in T_{min}$, then there must exist an edge $e'\in T_{min}$ that is
267 contained in~$C$ (take any pair of vertices separated by~$C$: the path
268 in~$T_{min}$ joining these vertices must cross~$C$ at least once). Exchanging
269 $e$ for $e'$ in $T_{min}$ yields an even lighter spanning tree since
273 \lemman{Red lemma, also known as the Cycle rule}\id{redlemma}%
274 An~edge~$e$ is not contained in the MST iff it is the heaviest on some cycle.
277 The implication from the left to the right follows directly from the Minimality
278 theorem: if~$e\not\in T_{min}$, then $e$~is $T_{min}$-heavy and so it is the heaviest
279 edge on the cycle $T_{min}[e]+e$.
281 We will prove the other implication again by contradiction. Suppose that $e$ is the heaviest edge of
282 a cycle~$C$ and that $e\in T_{min}$. Removing $e$ causes $T_{min}$ to split
283 to two components, let us call them $T_x$ and~$T_y$. Some vertices of~$C$ now lie in~$T_x$, the
284 others in~$T_y$, so there must exist in edge $e'\ne e$ such that its endpoints lie in different
285 components. Since $w(e')<w(e)$, exchanging $e$ for~$e'$ yields a~spanning tree lighter than
289 \figure{mst-rb.eps}{289pt}{Proof of the Blue (left) and Red (right) lemma}
291 \lemman{Black lemma}%
292 As long as there exists a black edge, at least one rule can be applied.
295 Assume that $e=xy$ be a black edge. Let us denote $M$ the set of vertices
296 reachable from~$x$ using only blue edges. If $y$~lies in~$M$, then $e$ together
297 with some blue path between $x$ and $y$ forms a cycle and it must be the heaviest
298 edge on this cycle. This holds because all blue edges have been already proven
299 to be in $T_{min}$ and there can be no $T_{min}$-light edges (see Theorem~\ref{mstthm}).
300 In this case we can apply the Red rule.
302 On the other hand, if $y\not\in M$, then the cut formed by all edges between $M$
303 and $V(G)\setminus M$ contains no blue edges, therefore we can use the Blue rule.
306 \figure{mst-bez.eps}{295pt}{Configurations in the proof of the Black lemma}
308 \thmn{Red-Blue correctness}%
309 For any selection of rules, the Red-Blue procedure stops and the blue edges form
310 the minimum spanning tree of the input graph.
313 To prove that the procedure stops, let us notice that no edge is ever recolored,
314 so we must run out of black edges after at most~$m$ steps. Recoloring
315 to the same color is avoided by the conditions built in the rules, recoloring to
316 a different color would mean that the an edge would be both inside and outside~$T_{min}$
317 due to our Red and Blue lemmata.
319 When no further rules can be applied, the Black lemma guarantees that all edges
320 are colored, so by the Blue lemma all blue edges are in~$T_{min}$ and by the Red
321 lemma all other (red) edges are outside~$T_{min}$, so the blue edges are exactly~$T_{min}$.
325 The MST problem is a~special case of the problem of finding the minimum basis
326 of a~weighted matroid. Surprisingly, when we modify the Red-Blue procedure to
327 use the standard definitions of cycles and cuts in matroids, it will always
328 find the minimum basis. Some of the other MST algorithms also easily generalize to
329 matroids and in some sense matroids are exactly the objects where ``the greedy approach works''. We
330 will however not pursue this direction in our work, referring the reader to the Oxley's monograph
331 \cite{oxley:matroids} instead.
333 %--------------------------------------------------------------------------------
335 \section{Classical algorithms}\id{classalg}%
337 The three classical MST algorithms can be easily stated in terms of the Red-Blue meta-algorithm.
338 For each of them, we first show the general version of the algorithm, then we prove that
339 it gives the correct result and finally we discuss the time complexity of various
342 \paran{Bor\o{u}vka's algorithm}%
343 The oldest MST algorithm is based on a~simple idea: grow a~forest in a~sequence of
344 iterations until it becomes connected. We start with a~forest of isolated
345 vertices. In each iteration, we let each tree of the forest select the lightest
346 edge of those having exactly one endpoint in the tree (we will call such edges
347 the \df{neighboring edges} of the tree). We add all such edges to the forest and
348 proceed with the next iteration.
350 \algn{Bor\o{u}vka \cite{boruvka:ojistem}, Choquet \cite{choquet:mst}, Sollin \cite{sollin:mst} and others}
352 \algin A~graph~$G$ with an edge comparison oracle.
353 \:$T\=$ a forest consisting of vertices of~$G$ and no edges.
354 \:While $T$ is not connected:
355 \::For each component $T_i$ of~$T$, choose the lightest edge $e_i$ from the cut
356 separating $T_i$ from the rest of~$T$.
357 \::Add all $e_i$'s to~$T$.
358 \algout Minimum spanning tree~$T$.
361 \lemma\id{boruvkadrop}%
362 In each iteration of the algorithm, the number of trees in~$T$ drops at least twice.
365 Each tree gets merged with at least one of its neighbors, so each of the new trees
366 contains two or more original trees.
370 The algorithm stops in $\O(\log n)$ iterations.
373 Bor\o{u}vka's algorithm outputs the MST of the input graph.
376 In every iteration of the algorithm, $T$ is a blue subgraph,
377 because every addition of some edge~$e_i$ to~$T$ is a straightforward
378 application of the Blue rule. We stop when the blue subgraph is connected, so
379 we do not need the Red rule to explicitly exclude edges.
381 It remains to show that adding the edges simultaneously does not
382 produce a cycle. Consider the first iteration of the algorithm where $T$ contains a~cycle~$C$. Without
383 loss of generality we can assume that $C=T_1[u_1v_1]\,v_1u_2\,T_2[u_2v_2]\,v_2u_3\,T_3[u_3v_3]\, \ldots \,T_k[u_kv_k]\,v_ku_1$.
384 Each component $T_i$ has chosen its lightest incident edge~$e_i$ as either the edge $v_iu_{i+1}$
385 or $v_{i-1}u_i$ (indexing cyclically). Suppose that $e_1=v_1u_2$ (otherwise we reverse the orientation
386 of the cycle). Then $e_2=v_2u_3$ and $w(e_2)<w(e_1)$ and we can continue in the same way,
387 getting $w(e_1)>w(e_2)>\ldots>w(e_k)>w(e_1)$, which is a contradiction.
388 (Note that distinctness of edge weights was crucial here.)
391 \lemma\id{boruvkaiter}%
392 Each iteration can be carried out in time $\O(m)$.
395 We assign a label to each tree and we keep a mapping from vertices to the
396 labels of the trees they belong to. We scan all edges, map their endpoints
397 to the particular trees and for each tree we maintain the lightest incident edge
398 so far encountered. Instead of merging the trees one by one (which would be too
399 slow), we build an auxilliary graph whose vertices are the labels of the original
400 trees and edges correspond to the chosen lightest inter-tree edges. We find connected
401 components of this graph, these determine how are the original labels translated
406 Bor\o{u}vka's algorithm finds the MST in time $\O(m\log n)$.
409 Follows from the previous lemmata.
412 \paran{Jarn\'\i{}k's algorithm}%
413 The next algorithm, discovered independently by Jarn\'\i{}k, Prim and Dijkstra, is similar
414 to Bor\o{u}vka's algorithm, but instead of the whole forest it concentrates on
415 a~single tree. It starts with a~single vertex and it repeatedly extends the tree
416 by the lightest neighboring edge until it spans the whole graph.
418 \algn{Jarn\'\i{}k \cite{jarnik:ojistem}, Prim \cite{prim:mst}, Dijkstra \cite{dijkstra:mst}}\id{jarnik}%
420 \algin A~graph~$G$ with an edge comparison oracle.
421 \:$T\=$ a single-vertex tree containing an~arbitrary vertex of~$G$.
422 \:While there are vertices outside $T$:
423 \::Pick the lightest edge $uv$ such that $u\in V(T)$ and $v\not\in V(T)$.
425 \algout Minimum spanning tree~$T$.
429 Jarn\'\i{}k's algorithm computes the MST of the input graph.
432 If~$G$ is connected, the algorithm always stops. Let us prove that in every step of
433 the algorithm, $T$ is always a blue tree. Step~4 corresponds to applying
434 the Blue rule to the cut $\delta(T)$ separating~$T$ from the rest of the given graph. We need not care about
435 the remaining edges, since for a connected graph the algorithm always stops with the right
436 number of blue edges.
440 The most important part of the algorithm is finding \em{neighboring edges.}
441 In a~straightforward implementation, searching for the lightest neighboring
442 edge takes $\Theta(m)$ time, so the whole algorithm runs in time $\Theta(mn)$.
444 We can do much better by using a binary
445 heap to hold all neighboring edges. In each iteration, we find and delete the
446 minimum edge from the heap and once we expand the tree, we insert the newly discovered
447 neighboring edges to the heap while deleting the neighboring edges that become
448 internal to the new tree. Since there are always at most~$m$ edges in the heap,
449 each heap operation takes $\O(\log m)=\O(\log n)$ time. For every edge, we perform
450 at most one insertion and at most one deletion, so we spend $\O(m\log n)$ time in total.
451 From this, we can conclude:
454 Jarn\'\i{}k's algorithm finds the MST of a~given graph in time $\O(m\log n)$.
457 We will show several faster implementations in section \ref{iteralg}.
459 \paran{Kruskal's algorithm}%
460 The last of the three classical algorithms processes the edges of the
461 graph~$G$ greedily. It starts with an~empty forest and it takes the edges of~$G$
462 in order of their increasing weights. For every edge, it checks whether its
463 addition to the forest produces a~cycle and if it does not, the edge is added.
464 Otherwise, the edge is dropped and not considered again.
466 \algn{Kruskal \cite{kruskal:mst}}
468 \algin A~graph~$G$ with an edge comparison oracle.
469 \:Sort edges of~$G$ by their increasing weights.
470 \:$T\=\emptyset$. \cmt{an empty spanning subgraph}
471 \:For all edges $e$ in their sorted order:
472 \::If $T+e$ is acyclic, add~$e$ to~$T$.
473 \::Otherwise drop~$e$.
474 \algout Minimum spanning tree~$T$.
478 Kruskal's algorithm returns the MST of the input graph.
481 In every step, $T$ is a forest of blue trees. Adding~$e$ to~$T$
482 in step~4 applies the Blue rule on the cut separating some pair of components of~$T$ ($e$ is the lightest,
483 because all other edges of the cut have not been considered yet). Dropping~$e$ in step~5 corresponds
484 to the Red rule on the cycle found ($e$~must be the heaviest, since all other edges of the
485 cycle have been already processed). At the end of the algorithm, all edges are colored,
486 so~$T$ must be the~MST.
490 Except for the initial sorting, which in general takes $\Theta(m\log m)$ time, the only
491 other non-trivial operation is the detection of cycles. What we need is a~data structure
492 for maintaining connected components, which supports queries and edge insertion.
493 This is closely related to the well-known Disjoint Set Union problem:
495 \problemn{Disjoint Set Union (DSU)}
496 Maintain an~equivalence relation on a~finite set under a~sequence of operations \<Union>
497 and \<Find>. The \<Find> operation tests whether two elements are equivalent and \<Union>
498 joins two different equivalence classes into one.
501 We can maintain the connected components of our forest~$T$ as equivalence classes. When we want
502 to add an~edge~$uv$, we first call $\<Find>(u,v)$ to check if both endpoints of the edge lie in
503 the same components. If they do not, addition of this edge connects both components into one,
504 so we perform $\<Union>(u,v)$ to merge the equivalence classes.
506 Tarjan and van Leeuwen have shown that there is a~data structure for the DSU problem
507 with surprising efficiency:
509 \thmn{Disjoint Set Union, Tarjan and van Leeuwen \cite{tarjan:setunion}}\id{dfu}%
510 Starting with a~trivial equivalence with single-element classes, a~sequence of operations
511 comprising of $n$~\<Union>s intermixed with $m\ge n$~\<Find>s can be processed in time
512 $\O(m\timesalpha(m,n))$, where $\alpha(m,n)$ is a~certain inverse of the Ackermann's function
513 (see Definition \ref{ackerinv}).
516 See \cite{tarjan:setunion}.
519 This completes the following theorem:
522 Kruskal's algorithm finds the MST of a given graph in time $\O(m\log n)$.
523 If the edges are already sorted by their weights, the time drops to
524 $\O(m\timesalpha(m,n))$.
527 We spend $\O(m\log n)$ on sorting, $\O(m\timesalpha(m,n))$ on processing the sequence
528 of \<Union>s and \<Find>s, and $\O(m)$ on all other work.
532 The cost of the \<Union> and \<Find> operations is of course dwarfed by the complexity
533 of sorting, so a much simpler (at least in terms of its analysis) data
534 structure would be sufficient, as long as it has $\O(\log n)$ amortized complexity
535 per operation. For example, we can label vertices with identifiers of the
536 corresponding components and always relabel the smaller of the two components.
538 We will study dynamic maintenance of connected components in more detail in Chapter~\ref{dynchap}.
540 %--------------------------------------------------------------------------------
542 \section{Contractive algorithms}\id{contalg}%
544 While the classical algorithms are based on growing suitable trees, they
545 can be also reformulated in terms of edge contraction. Instead of keeping
546 a forest of trees, we can keep each tree contracted to a single vertex.
547 This replaces the relatively complex tree-edge incidencies by simple
548 vertex-edge incidencies, potentially speeding up the calculation at the
549 expense of having to perform the contractions.
551 We will show a contractive version of the Bor\o{u}vka's algorithm
552 in which these costs are carefully balanced, leading for example to
553 a linear-time algorithm for MST in planar graphs.
555 There are two definitions of edge contraction that differ when an edge of
556 a~triangle is contracted. Either we unify the other two edges to a single edge
557 or we keep them as two parallel edges, leaving us with a~multigraph. We will
558 use the multigraph version and we will show that we can easily reduce the multigraph
559 to a simple graph later. (See \ref{contract} for the exact definitions.)
561 We only need to be able to map edges of the contracted graph to the original
562 edges, so each edge will carry a unique label $\ell(e)$ that will be preserved by
565 \lemman{Flattening a multigraph}\id{flattening}%
566 Let $G$ be a multigraph and $G'$ its subgraph such that all loops have been
567 removed and each bundle of parallel edges replaced by its lightest edge.
568 Then $G'$~has the same MST as~$G$.
571 Every spanning tree of~$G'$ is a spanning tree of~$G$. In the other direction:
572 Loops can be never contained in a spanning tree. If there is a spanning tree~$T$
573 containing a~removed edge~$e$ parallel to an edge~$e'\in G'$, exchaning $e'$
574 for~$e$ makes~$T$ lighter. \qed
576 \rem Removal of the heavier of a pair of parallel edges can be also viewed
577 as an application of the Red rule on a two-edge cycle. And indeed it is, the
578 Red-Blue procedure works on multigraphs as well as on simple graphs and all the
579 classical algorithms also do. We would only have to be more careful in the
580 formulations and proofs, which we preferred to avoid.
582 \algn{Contractive version of Bor\o{u}vka's algorithm}\id{contbor}
584 \algin A~graph~$G$ with an edge comparison oracle.
586 \:$\ell(e)\=e$ for all edges~$e$. \cmt{Initialize the labels.}
588 \::For each vertex $v_k$ of~$G$, let $e_k$ be the lightest edge incident to~$v_k$.
589 \::$T\=T\cup \{ \ell(e_k) \}$. \cmt{Remember labels of all selected edges.}
590 \::Contract $G$ along all edges $e_k$, inheriting labels and weights.\foot{In other words, we ask the comparison oracle for the edge $\ell(e)$ instead of~$e$.}
591 \::Flatten $G$, removing parallel edges and loops.
592 \algout Minimum spanning tree~$T$.
596 For the analysis of the algorithm, we will denote the graph considered by the algorithm
597 at the beginning of the $i$-th iteration by $G_i$ (starting with $G_0=G$) and the number
598 of vertices and edges of this graph by $n_i$ and $m_i$ respectively.
601 The $i$-th iteration of the algorithm (also called the \df{Bor\o{u}vka step}) can be carried
602 out in time~$\O(m_i)$.
605 The only non-trivial parts are steps 6 and~7. Contractions can be handled similarly
606 to the unions in the original Bor\o{u}vka's algorithm (see \ref{boruvkaiter}):
607 We build an auxillary graph containing only the selected edges~$e_k$, find
608 connected components of this graph and renumber vertices in each component to
609 the identifier of the component. This takes $\O(m_i)$ time.
611 Flattening is performed by first removing the loops and then bucket-sorting the edges
612 (as ordered pairs of vertex identifiers) lexicographically, which brings parallel
613 edges together. The bucket sort uses two passes with $n_i$~buckets, so it takes
614 $\O(n_i+m_i)=\O(m_i)$.
618 The Contractive Bor\o{u}vka's algorithm finds the MST of the input graph in
619 time $\O(\min(n^2,m\log n))$.
622 As in the original Bor\o{u}vka's algorithm, the number of iterations is $\O(\log n)$.
623 When combined with the previous lemma, it gives an~$\O(m\log n)$ upper bound.
625 To get the $\O(n^2)$ bound, we observe that the number of trees in the non-contracting
626 version of the algorithm drops at least by a factor of two in each iteration (Lemma \ref{boruvkadrop})
627 and the same must hold for the number of vertices in the contracting version.
628 Therefore $n_i\le n/2^i$. While the number of edges need not decrease geometrically,
629 we still have $m_i\le n_i^2$ as the graphs~$G_i$ are simple (we explicitly removed multiple
630 edges and loops at the end of the previous iteration). Hence the total time spent
631 in all iterations is $\O(\sum_i n_i^2) = \O(\sum_i n^2/4^i) = \O(n^2)$.
634 \thmn{Contractive Bor\o{u}vka on planar graphs, \cite{mm:mst}}\id{planarbor}%
635 When the input graph is planar, the Contractive Bor\o{u}vka's algorithm runs in
639 Let us refine the previous proof. We already know that $n_i \le n/2^i$. We will
640 prove that when~$G$ is planar, the $m_i$'s are decreasing geometrically. We know that every
641 $G_i$ is planar, because the class of planar graphs is closed under edge deletion and
642 contraction. Moreover, $G_i$~is also simple, so we can use the standard theorem on
643 the number of edges of planar simple graphs (see for example \cite{diestel:gt}) to get $m_i\le 3n_i \le 3n/2^i$.
644 The total time complexity of the algorithm is therefore $\O(\sum_i m_i)=\O(\sum_i n/2^i)=\O(n)$.
648 There are several other possibilities how to find the MST of a planar graph in linear time.
649 For example, Matsui \cite{matsui:planar} has described an algorithm based on simultaneously
650 working on the graph and its topological dual. The advantage of our approach is that we do not need
651 to construct the planar embedding explicitly. We will show one more linear algorithm
652 in section~\ref{minorclosed}.
655 To achieve the linear time complexity, the algorithm needs a very careful implementation,
656 but we defer the technical details to section~\ref{bucketsort}.
658 \paran{General contractions}%
659 Graph contractions are indeed a~very powerful tool and they can be used in other MST
660 algorithms as well. The following lemma shows the gist:
662 \lemman{Contraction of MST edges}\id{contlemma}%
663 Let $G$ be a weighted graph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
664 produced by contracting $G$ along~$e$, and $\pi$ the bijection between edges of~$G-e$ and
665 their counterparts in~$G/e$. Then: $$\mst(G) = \pi^{-1}[\mst(G/e)] + e.$$
668 % We seem not to need this lemma for multigraphs...
669 %If there are any loops or parallel edges in~$G$, we can flatten the graph. According to the
670 %Flattening lemma (\ref{flattening}), the MST stays the same and if we remove a parallel edge
671 %or loop~$f$, then $\pi(f)$ would be removed when flattening~$G/e$, so $f$ never participates
673 The right-hand side of the equality is a spanning tree of~$G$, let us denote it by~$T$ and
674 the MST of $G/e$ by~$T'$. If $T$ were not minimum, there would exist a $T$-light edge~$f$ in~$G$
675 (by Theorem \ref{mstthm}). If the path $T[f]$ covered by~$f$ does not contain~$e$,
676 then $\pi[T[f]]$ is a path covered by~$\pi(f)$ in~$T'$. Otherwise $\pi(T[f]-e)$ is such a path.
677 In both cases, $f$ is $T'$-light, which contradicts the minimality of~$T'$. (We do not have
678 a~multigraph version of the theorem, but the side we need is a~straightforward edge exchange,
679 which obviously works in multigraphs as well.)
683 In the previous algorithm, the role of the mapping~$\pi^{-1}$ is of course played by the edge labels~$\ell$.
685 \paran{A~lower bound}%
686 Finally, we will show a family of graphs where the $\O(m\log n)$ bound on time complexity
687 is tight. The graphs do not have unique weights, but they are constructed in a way that
688 the algorithm never compares two edges with the same weight. Therefore, when two such
689 graphs are monotonely isomorphic (see~\ref{mstiso}), the algorithm processes them in the same way.
692 A~\df{distractor of order~$k$,} denoted by~$D_k$, is a path on $n=2^k$~vertices $v_1,\ldots,v_n$
693 where each edge $v_iv_{i+1}$ has its weight equal to the number of trailing zeroes in the binary
694 representation of the number~$i$. The vertex $v_1$ is called a~\df{base} of the distractor.
697 Alternatively, we can use a recursive definition: $D_0$ is a single vertex, $D_{k+1}$ consists
698 of two disjoint copies of~$D_k$ joined by an edge of weight~$k$.
700 \figure{distractor.eps}{\epsfxsize}{A~distractor $D_3$ and its evolution (bold edges are contracted)}
703 A~single iteration of the contractive algorithm reduces~$D_k$ to a graph isomorphic with~$D_{k-1}$.
706 Each vertex~$v$ of~$D_k$ is incident with a single edge of weight~1. The algorithm therefore
707 selects all weight~1 edges and contracts them. This produces a graph which is
708 exactly $D_{k-1}$ with all weights increased by~1, which does not change the relative order of edges.
712 A~\df{hedgehog}~$H_{a,k}$ is a graph consisting of $a$~distractors $D_k^1,\ldots,D_k^a$ of order~$k$
713 together with edges of a complete graph on the bases of the distractors. These additional edges
714 have arbitrary weights, but heavier than the edges of all distractors.
716 \figure{hedgehog.eps}{\epsfxsize}{A~hedgehog $H_{5,2}$ (quills bent to fit in the picture)}
719 A~single iteration of the contractive algorithm reduces~$H_{a,k}$ to a graph isomorphic with $H_{a,k-1}$.
722 Each vertex is incident with an edge of some distractor, so the algorithm does not select
723 any edge of the complete graph. Contraction therefore reduces each distractor to a smaller
724 distractor (modulo an additive factor in weight) and leaves the complete graph intact.
725 This is monotonely isomorphic to $H_{a,k-1}$.
728 \thmn{Lower bound for Contractive Bor\o{u}vka}%
729 For each $n$ there exists a graph on $\Theta(n)$ vertices and $\Theta(n)$ edges
730 such that the Contractive Bor\o{u}vka's algorithm spends time $\Omega(n\log n)$ on it.
733 Consider the hedgehog $H_{a,k}$ for $a=\lceil\sqrt n\rceil$ and $k=\lceil\log_2 a\rceil$.
734 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
737 By the previous lemma, the algorithm proceeds through a sequence of hedgehogs $H_{a,k},
738 H_{a,k-1}, \ldots, H_{a,0}$ (up to monotone isomorphism), so it needs a logarithmic number of iterations plus some more
739 to finish on the remaining complete graph. Each iteration runs on a graph with $\Omega(n)$
740 edges as every $H_{a,k}$ contains a complete graph on~$a$ vertices.
743 \remn{Disconnected graphs}\id{disconn}%
744 The basic properties of minimum spanning trees and the algorithms presented in
745 this chapter apply to minimum spanning forests of disconnected graphs, too.
746 The proofs of our theorems and the steps of our algorithms are based on adjacency
747 of vertices and existence of paths, so they are always local to a~single
748 connected component. The Bor\o{u}vka's and Kruskal's algorithm need no changes,
749 the Jarn\'\i{}k's algorithm has to be invoked separately for each component.
751 We can also extend the notion of light and heavy edges with respect
752 to a~tree to forests: When an~edge~$e$ connects two vertices lying in the same
753 tree~$T$ of a~forest~$F$, it is $F$-heavy iff it is $T$-heavy (similarly
754 for $F$-light). Edges connecting two different trees are always considered
755 $F$-light. Again, a~spanning forest~$F$ is minimum iff there are no $F$-light