5 \chapter{Advanced MST Algorithms}
7 \section{Minor-closed graph classes}\id{minorclosed}%
9 The contractive algorithm given in section~\ref{contalg} has been found to perform
10 well on planar graphs, but in the general case its time complexity was not linear.
11 Can we find any broader class of graphs where the algorithm is still efficient?
12 The right context turns out to be the minor-closed graph classes, which are
13 closed under contractions and have bounded density.
16 A~graph~$H$ is a \df{minor} of a~graph~$G$ (written as $H\minorof G$) iff it can be obtained
17 from a~subgraph of~$G$ by a sequence of simple graph contractions (see \ref{simpcont}).
20 A~class~$\cal C$ of graphs is \df{minor-closed}, when for every $G\in\cal C$ and
21 its every minor~$H$, the graph~$H$ lies in~$\cal C$ as well. A~class~$\cal C$ is called
22 \df{non-trivial} if at least one graph lies in~$\cal C$ and at least one lies outside~$\cal C$.
25 Non-trivial minor-closed classes include:
28 \:graphs embeddable in any fixed surface (i.e., graphs of bounded genus),
29 \:graphs embeddable in~${\bb R}^3$ without knots or without interlocking cycles,
30 \:graphs of bounded tree-width or path-width.
34 Many of the nice structural properties of planar graphs extend to
35 minor-closed classes, too (see \cite{diestel:gt} for a~nice overview
36 of this theory). The most important property is probably the characterization
37 of such classes by forbidden minors.
40 For a~class~$\cal H$ of graphs we define $\Forb({\cal H})$ as the class
41 of graphs which do not contain any of the graphs in~$\cal H$ as a~minor.
42 We will call $\cal H$ the set of \df{forbidden (or excluded) minors} for this class.
45 For every~${\cal H}\ne\emptyset$, the class $\Forb({\cal H})$ is non-trivial
46 and closed on minors. This works in the opposite direction as well: for every
47 minor-closed class~$\cal C$ there is a~class $\cal H$ such that ${\cal C}=\Forb({\cal H})$.
48 One such~$\cal H$ is the complement of~$\cal C$, but smaller sets can be found, too.
49 For example, the planar graphs exclude exactly $K_5$ and~$K_{3,3}$ --- this follows
50 from the Kuratowski's theorem (the theorem uses forbidden subdivisions, but while
51 in general this is not the same as forbidden minors, it is for $K_5$ and $K_{3,3}$).
52 The celebrated theorem by Robertson and Seymour guarantees that we can always find
53 a~finite set of forbidden minors.
55 \thmn{Excluded minors, Robertson \& Seymour \cite{rs:wagner}}
56 For every non-trivial minor-closed graph class~$\cal C$ there exists
57 a~finite set~$\cal H$ of graphs such that ${\cal C}=\Forb({\cal H})$.
60 This theorem has been proven in the series of papers on graph minors
61 culminating with~\cite{rs:wagner}. See this paper and follow the references
62 to the previous articles in the series.
66 We will make use of another important property --- the bounded density of minor-closed
67 classes. The connection between minors and density dates back to Mader in the 1960's
68 and it can be proven without use of the Robertson-Seymour theorem.
71 Let $\cal C$ be a class of graphs. We define its \df{edge density} $\varrho(\cal C)$
72 to be the infimum of all~$\varrho$'s such that $m(G) \le \varrho\cdot n(G)$
73 holds for every $G\in\cal C$.
75 \thmn{Mader \cite{mader:dens}}
76 For every $k\in{\bb N}$ there exists $h(k)\in{\bb R}$ such that every graph
77 with average degree at least~$h(k)$ contains a~subdivision of~$K_{k}$ as a~subgraph.
80 (See Lemma 3.5.1 in \cite{diestel:gt} for a~complete proof in English.)
82 Let us fix~$k$ and prove by induction on~$m$ that every graph with average
83 degree $a\ge 2^m$ contains a~subdivision of some graph with $k$~vertices
84 and ${k\choose 2}\ge m\ge k$~edges. For $m={k\choose 2}$ the theorem follows
85 as the only graph with~$k$ vertices and~$k\choose 2$ edges is~$K_k$.
87 The base case $m=k$: Let us observe that when the average degree
88 is~$a$, removing any vertex of degree less than~$a/2$ does not decrease the
89 average degree. A~graph with $a\ge 2^k$ therefore has a~subgraph
90 with minimum degree $\delta\ge a/2=2^{k-1}$. Such subgraph contains
91 a~cycle on more than~$\delta$ vertices, in other words a~subdivision of
94 Induction step: Let~$G$ be a~graph with average degree at least~$2^m$ and
95 assume that the theorem already holds for $m-1$. Without loss of generality,
96 $G$~is connected. Consider a~maximal set $U\subseteq V$ such that $G[U]$ is connected
97 and the graph $G.U$ ($G$~with $U$~contracted to a~single vertex) has average
98 degree at least~$2^m$ (such~$U$ exists, because $G=G.U$ whenever $\vert U\vert=1$).
99 Now consider the subgraph~$H$ induced in~$G$ by the
100 neighbors of~$U$. Every $v\in V(H)$ must have degree at least~$2^{m-1}$
101 (otherwise we can add this vertex to~$U$, contradicting its maximality), so by
102 the induction hypothesis $H$ contains a~subdivision of some graph~$R$ with
103 $r$~vertices and $m-1$ edges. Any two non-adjacent vertices of~$R$ can be
104 connected in the subdivision by a~path lying entirely in~$G[U]$, which reveals
105 a~subdivision of a~graph on $m$~vertices. \qed
107 \thmn{Density of minor-closed classes, Mader~\cite{mader:dens}}
108 Every non-trivial minor-closed class of graphs has finite edge density.
111 Let~$\cal C$ be any such class, $X$~its smallest excluded minor and $x=n(X)$.
112 As $H\minorof K_x$, the class $\cal C$ entirely lies in ${\cal C}'=\Forb(\{K_x\})$, so
113 $\varrho({\cal C}) \le \varrho({\cal C}')$ and therefore it suffices to prove the
114 theorem for classes excluding a~single complete graph~$K_x$.
116 We will show that $\varrho({\cal C})\le 2h(x)$, where $h$~is the function
117 from the previous theorem. If any $G\in{\cal C}$ had more than $2h(x)\cdot n(G)$
118 edges, its average degree would be at least~$h(x)$, so by the previous theorem
119 $G$~would contain a~subdivision of~$K_x$ and therefore $K_x$ as a~minor.
123 See also Theorem 6.1 in \cite{nesetril:minors}, which also lists some other equivalent conditions.
125 \thmn{MST on minor-closed classes \cite{mm:mst}}\id{mstmcc}%
126 For any fixed non-trivial minor-closed class~$\cal C$ of graphs, Algorithm \ref{contbor} finds
127 the MST of any graph in this class in time $\O(n)$. (The constant hidden in the~$\O$
128 depends on the class.)
131 Following the proof for planar graphs (\ref{planarbor}), we denote the graph considered
132 by the algorithm at the beginning of the $i$-th iteration by~$G_i$ and its number of vertices
133 and edges by $n_i$ and $m_i$ respectively. Again the $i$-th phase runs in time $\O(m_i)$
134 and $n_i \le n/2^i$, so it remains to show a linear bound for the $m_i$'s.
136 Since each $G_i$ is produced from~$G_{i-1}$ by a sequence of edge contractions,
137 all $G_i$'s are minors of~$G$.\foot{Technically, these are multigraph contractions,
138 but followed by flattening, so they are equivalent to contractions on simple graphs.}
139 So they also belong to~$\cal C$ and by the previous theorem $m_i\le \varrho({\cal C})\cdot n_i$.
143 The contractive algorithm uses ``batch processing'' to perform many contractions
144 in a single step. It is also possible to perform contractions one edge at a~time,
145 batching only the flattenings. A~contraction of an edge~$uv$ can be done
146 in time~$\O(\deg(u))$ by removing all edges incident with~$u$ and inserting them back
147 with $u$ replaced by~$v$. Therefore we need to find a lot of vertices with small
148 degrees. The following lemma shows that this is always the case in minor-closed
151 \lemman{Low-degree vertices}\id{lowdeg}%
152 Let $\cal C$ be a graph class with density~$\varrho$ and $G\in\cal C$ a~graph
153 with $n$~vertices. Then at least $n/2$ vertices of~$G$ have degree at most~$4\varrho$.
156 Assume the contrary: Let there be at least $n/2$ vertices with degree
157 greater than~$4\varrho$. Then $\sum_v \deg(v) > n/2
158 \cdot 4\varrho = 2\varrho n$, which is in contradiction with the number
159 of edges being at most $\varrho n$.
163 The proof can be also viewed
164 probabilistically: let $X$ be the degree of a vertex of~$G$ chosen uniformly at
165 random. Then ${\bb E}X \le 2\varrho$, hence by the Markov's inequality
166 ${\rm Pr}[X > 4\varrho] < 1/2$, so for at least $n/2$ vertices~$v$ we have
167 $\deg(v)\le 4\varrho$.
169 \algn{Local Bor\o{u}vka's Algorithm \cite{mm:mst}}%
171 \algin A~graph~$G$ with an edge comparison oracle and a~parameter~$t\in{\bb N}$.
173 \:$\ell(e)\=e$ for all edges~$e$.
175 \::While there exists a~vertex~$v$ such that $\deg(v)\le t$:
176 \:::Select the lightest edge~$e$ incident with~$v$.
177 \:::Contract~$G$ along~$e$.
178 \:::$T\=T + \ell(e)$.
179 \::Flatten $G$, removing parallel edges and loops.
180 \algout Minimum spanning tree~$T$.
184 When $\cal C$ is a minor-closed class of graphs with density~$\varrho$, the
185 Local Bor\o{u}vka's Algorithm with the parameter~$t$ set to~$4\varrho$
186 finds the MST of any graph from this class in time $\O(n)$. (The constant
187 in the~$\O$ depends on~the class.)
190 Let us denote by $G_i$, $n_i$ and $m_i$ the graph considered by the
191 algorithm at the beginning of the $i$-th iteration of the outer loop,
192 and the number of its vertices and edges respectively. As in the proof
193 of the previous algorithm (\ref{mstmcc}), we observe that all the $G_i$'s
194 are minors of the graph~$G$ given as the input.
196 For the choice $t=4\varrho$, the Lemma on low-degree vertices (\ref{lowdeg})
197 guarantees that at least $n_i/2$ edges get selected in the $i$-th iteration.
198 Hence at least a half of the vertices participates in contractions, so
199 $n_i\le 3/4\cdot n_{i-1}$. Therefore $n_i\le n\cdot (3/4)^i$ and the algorithm terminates
200 after $\O(\log n)$ iterations.
202 Each selected edge belongs to $\mst(G)$, because it is the lightest edge of
203 the trivial cut $\delta(v)$ (see the Blue Rule in \ref{rbma}).
204 The steps 6 and~7 therefore correspond to the operation
205 described by the Lemma on contraction of MST edges (\ref{contlemma}) and when
206 the algorithm stops, $T$~is indeed the minimum spanning tree.
208 It remains to analyse the time complexity of the algorithm. Since $G_i\in{\cal C}$, we have
209 $m_i\le \varrho n_i \le \varrho n/2^i$.
210 We will show that the $i$-th iteration is carried out in time $\O(m_i)$.
211 Steps 5 and~6 run in time $\O(\deg(v))=\O(t)$ for each~$v$, so summed
212 over all $v$'s they take $\O(tn_i)$, which is linear for a fixed class~$\cal C$.
213 Flattening takes $\O(m_i)$, as already noted in the analysis of the Contracting
214 Bor\o{u}vka's Algorithm (see \ref{contiter}).
216 The whole algorithm therefore runs in time $\O(\sum_i m_i) = \O(\sum_i n/2^i) = \O(n)$.
220 For planar graphs, we can get a sharper version of the low-degree lemma,
221 showing that the algorithm works with $t=8$ as well (we had $t=12$ as
222 $\varrho=3$). While this does not change the asymptotic time complexity
223 of the algorithm, the constant-factor speedup can still delight the hearts of
226 \lemman{Low-degree vertices in planar graphs}%
227 Let $G$ be a planar graph with $n$~vertices. Then at least $n/2$ vertices of~$v$
228 have degree at most~8.
231 It suffices to show that the lemma holds for triangulations (if there
232 are any edges missing, the situation can only get better) with at
233 least 3 vertices. Since $G$ is planar, $\sum_v \deg(v) < 6n$.
234 The numbers $d(v):=\deg(v)-3$ are non-negative and $\sum_v d(v) < 3n$,
235 so by the same argument as in the proof of the general lemma, for at least $n/2$
236 vertices~$v$ it holds that $d(v) < 6$, hence $\deg(v) \le 8$.
240 The constant~8 in the previous lemma is the best we can have.
241 Consider a $k\times k$ triangular grid. It has $n=k^2$ vertices, $\O(k)$ of them
242 lie on the outer face and have degrees at most~6, the remaining $n-\O(k)$ interior
243 vertices have degree exactly~6. Therefore the number of faces~$f$ is $6/3\cdot n=2n$,
244 ignoring terms of order $\O(k)$. All interior triangles can be properly colored with
245 two colors, black and white. Now add a~new vertex inside each white face and connect
246 it to all three vertices on the boundary of that face. This adds $f/2 \approx n$
247 vertices of degree~3 and it increases the degrees of the original $\approx n$ interior
248 vertices to~9, therefore about a half of the vertices of the new planar graph
251 \figure{hexangle.eps}{\epsfxsize}{The construction from Remark~\ref{hexa}}
254 The observation in~Theorem~\ref{mstmcc} was also made by Gustedt in~\cite{gustedt:parallel},
255 who studied a~parallel version of the contractive Bor\o{u}vka's algorithm applied
256 to minor-closed classes.
258 %--------------------------------------------------------------------------------
260 \section{Using Fibonacci heaps}
263 We have seen that the Jarn\'\i{}k's Algorithm \ref{jarnik} runs in $\Theta(m\log n)$ time.
264 Fredman and Tarjan have shown a~faster implementation in~\cite{ft:fibonacci}
265 using their Fibonacci heaps. In this section, we convey their results and we
266 show several interesting consequences.
268 The previous implementation of the algorithm used a binary heap to store all edges
269 separating the current tree~$T$ from the rest of the graph, i.e., edges of the cut~$\delta(T)$.
270 Instead of that, we will remember the vertices adjacent to~$T$ and for each such vertex~$v$ we
271 will maintain the lightest edge~$uv$ such that $u$~lies in~$T$. We will call these edges \df{active edges}
272 and keep them in a~Fibonacci heap, ordered by weight.
274 When we want to extend~$T$ by the lightest edge of~$\delta(T)$, it is sufficient to
275 find the lightest active edge~$uv$ and add this edge to~$T$ together with the new vertex~$v$.
276 Then we have to update the active edges as follows. The edge~$uv$ has just ceased to
277 be active. We scan all neighbors~$w$ of the vertex~$v$. When $w$~is in~$T$, no action
278 is needed. If $w$~is outside~$T$ and it was not adjacent to~$T$ (there is no active edge
279 remembered for it so far), we set the edge~$vw$ as active. Otherwise we check the existing
280 active edge for~$w$ and replace it by~$vw$ if the new edge is lighter.
282 The following algorithm shows how these operations translate to insertions, decreases
283 and deletions on the heap.
285 \algn{Active Edge Jarn\'\i{}k; Fredman and Tarjan \cite{ft:fibonacci}}\id{jarniktwo}%
287 \algin A~graph~$G$ with an edge comparison oracle.
288 \:$v_0\=$ an~arbitrary vertex of~$G$.
289 \:$T\=$ a tree containing just the vertex~$v_0$.
290 \:$H\=$ a~Fibonacci heap of active edges stored as pairs $(u,v)$ where $u\in T,v\not\in T$, ordered by the weights $w(uv)$, initially empty.
291 \:$A\=$ a~mapping of vertices outside~$T$ to their active edges in the heap; initially all elements undefined.
292 \:\<Insert> all edges incident with~$v_0$ to~$H$ and update~$A$ accordingly.
293 \:While $H$ is not empty:
294 \::$(u,v)\=\<DeleteMin>(H)$.
296 \::For all edges $vw$ such that $w\not\in T$:
297 \:::If there exists an~active edge~$A(w)$:
298 \::::If $vw$ is lighter than~$A(w)$, \<Decrease> $A(w)$ to~$(v,w)$ in~$H$.
299 \:::If there is no such edge, then \<Insert> $(v,w)$ to~$H$ and set~$A(w)$.
300 \algout Minimum spanning tree~$T$.
304 To analyze the time complexity of this algorithm, we will use the standard
305 theorem on~complexity of the Fibonacci heap:
307 \thmn{Fibonacci heaps} The~Fibonacci heap performs the following operations
308 with the indicated amortized time complexities:
310 \:\<Insert> (insertion of a~new element) in $\O(1)$,
311 \:\<Decrease> (decreasing value of an~existing element) in $\O(1)$,
312 \:\<Merge> (merging of two heaps into one) in $\O(1)$,
313 \:\<DeleteMin> (deletion of the minimal element) in $\O(\log n)$,
314 \:\<Delete> (deletion of an~arbitrary element) in $\O(\log n)$,
316 \>where $n$ is the number of elements present in the heap at the time of
320 See Fredman and Tarjan \cite{ft:fibonacci} for both the description of the Fibonacci
321 heap and the proof of this theorem.
325 Algorithm~\ref{jarniktwo} with the Fibonacci heap finds the MST of the input graph in time~$\O(m+n\log n)$.
328 The algorithm always stops, because every edge enters the heap~$H$ at most once.
329 As it selects exactly the same edges as the original Jarn\'\i{}k's algorithm,
330 it gives the correct answer.
332 The time complexity is $\O(m)$ plus the cost of the heap operations. The algorithm
333 performs at most one \<Insert> or \<Decrease> per edge and exactly one \<DeleteMin>
334 per vertex. There are at most $n$ elements in the heap at any given time,
335 thus by the previous theorem the operations take $\O(m+n\log n)$ time in total.
339 For graphs with edge density at least $\log n$, this algorithm runs in linear time.
342 We can consider using other kinds of heaps which have the property that inserts
343 and decreases are faster than deletes. Of course, the Fibonacci heaps are asymptotically
344 optimal (by the standard $\Omega(n\log n)$ lower bound on sorting by comparisons, see
345 for example \cite{clrs}), so the other data structures can improve only
346 multiplicative constants or offer an~easier implementation.
348 A~nice example is a~\df{$d$-regular heap} --- a~variant of the usual binary heap
349 in the form of a~complete $d$-regular tree. \<Insert>, \<Decrease> and other operations
350 involving bubbling the values up spend $\O(1)$ time at a~single level, so they run
351 in~$\O(\log_d n)$ time. \<Delete> and \<DeleteMin> require bubbling down, which incurs
352 comparison with all~$d$ sons at every level, so they spend $\O(d\log_d n)$.
353 With this structure, the time complexity of the whole algorithm
354 is $\O(nd\log_d n + m\log_d n)$, which suggests setting $d=m/n$, yielding $\O(m\log_{m/n}n)$.
355 This is still linear for graphs with density at~least~$n^{1+\varepsilon}$.
357 Another possibility is to use the 2-3-heaps \cite{takaoka:twothree} or Trinomial
358 heaps \cite{takaoka:trinomial}. Both have the same asymptotic complexity as Fibonacci
359 heaps (the latter even in the worst case, but it does not matter here) and their
360 authors claim faster implementation.
362 \FIXME{Mention Thorup's Fibonacci-like heaps for integers?}
365 As we already noted, the improved Jarn\'\i{}k's algorithm runs in linear time
366 for sufficiently dense graphs. In some cases, it is useful to combine it with
367 another MST algorithm, which identifies a~part of the MST edges and contracts
368 the graph to increase its density. For example, we can perform several
369 iterations of the Contractive Bor\o{u}vka's algorithm and find the rest of the
370 MST by the Active Edge Jarn\'\i{}k's algorithm.
372 \algn{Mixed Bor\o{u}vka-Jarn\'\i{}k}
374 \algin A~graph~$G$ with an edge comparison oracle.
375 \:Run $\log\log n$ iterations of the Contractive Bor\o{u}vka's algorithm (\ref{contbor}),
377 \:Run the Active Edge Jarn\'\i{}k's algorithm (\ref{jarniktwo}) on the resulting
378 graph, getting a~MST~$T_2$.
379 \:Combine $T_1$ and~$T_2$ to~$T$ as in the Contraction lemma (\ref{contlemma}).
380 \algout Minimum spanning tree~$T$.
384 The Mixed Bor\o{u}vka-Jarn\'\i{}k algorithm finds the MST of the input graph in time $\O(m\log\log n)$.
387 Correctness follows from the Contraction lemma and from the proofs of correctness of the respective algorithms.
388 As~for time complexity: The first step takes $\O(m\log\log n)$ time
389 (by Lemma~\ref{contiter}) and it gradually contracts~$G$ to a~graph~$G'$ of size
390 $m'\le m$ and $n'\le n/\log n$. The second step then runs in time $\O(m'+n'\log n') = \O(m)$
391 and both trees can be combined in linear time, too.
395 Actually, there is a~much better choice of the algorithms to combine: use the
396 Active Edge Jarn\'\i{}k's algorithm multiple times, each time stopping after a~while.
397 A~good choice of the stopping condition is to place a~limit on the size of the heap.
398 We start with an~arbitrary vertex, grow the tree as usually and once the heap gets too large,
399 we conserve the current tree and start with a~different vertex and an~empty heap. When this
400 process runs out of vertices, it has identified a~sub-forest of the MST, so we can
401 contract the graph along the edges of~this forest and iterate.
403 \algn{Iterated Jarn\'\i{}k; Fredman and Tarjan \cite{ft:fibonacci}}
405 \algin A~graph~$G$ with an edge comparison oracle.
406 \:$T\=\emptyset$. \cmt{edges of the MST}
407 \:$\ell(e)\=e$ for all edges~$e$. \cmt{edge labels as usually}
409 \:While $n>1$: \cmt{We will call iterations of this loop \df{phases}.}
410 \::$F\=\emptyset$. \cmt{forest built in the current phase}
411 \::$t\=2^{\lceil 2m_0/n \rceil}$. \cmt{the limit on heap size}
412 \::While there is a~vertex $v_0\not\in F$:
413 \:::Run the Active Edge Jarn\'\i{}k's algorithm (\ref{jarniktwo}) from~$v_0$, stop when:
414 \::::all vertices have been processed, or
415 \::::a~vertex of~$F$ has been added to the tree, or
416 \::::the heap has grown to more than~$t$ elements.
417 \:::Denote the resulting tree~$R$.
419 \::$T\=T\cup \ell[F]$. \cmt{Remember MST edges found in this phase.}
420 \::Contract~$G$ along all edges of~$F$ and flatten it.
421 \algout Minimum spanning tree~$T$.
425 For analysis of the algorithm, let us denote the graph entering the $i$-th
426 phase by~$G_i$ and likewise with the other parameters. Let the trees from which
427 $F_i$~has been constructed be called $R_i^1, \ldots, R_i^{z_i}$. The
428 non-indexed $G$, $m$ and~$n$ will correspond to the graph given as~input.
431 However the choice of the parameter~$t$ can seem mysterious, the following
432 lemma makes the reason clear:
435 The $i$-th phase of the Iterated Jarn\'\i{}k's algorithm runs in time~$\O(m)$.
438 During the phase, the heap always contains at most~$t_i$ elements, so it takes
439 time~$\O(\log t_i)=\O(m/n_i)$ to delete an~element from the heap. The trees~$R_i^j$
440 are edge-disjoint, so there are at most~$n_i$ \<DeleteMin>'s over the course of the phase.
441 Each edge is considered at most twice (once per its endpoint), so the number
442 of the other heap operations is~$\O(m_i)$. Together, it equals $\O(m_i + n_i\log t_i) = \O(m_i+m) = \O(m)$.
446 Unless the $i$-th phase is final, the forest~$F_i$ consists of at most $2m_i/t_i$ trees.
449 As every edge of~$G_i$ is incident with at most two trees of~$F_i$, it is sufficient
450 to establish that there are at least~$t_i$ edges incident with every such tree, including
451 connecting two vertices of the tree.
453 The forest~$F_i$ evolves by additions of the trees~$R_i^j$. Let us consider the possibilities
454 how the algorithm could have stopped growing the tree~$R_i^j$:
456 \:the heap had more than~$t_i$ elements (step~10): since the each elements stored in the heap
457 corresponds to a~unique edges incident with~$R_i^j$, we have enough such edges;
458 \:the algorithm just added a~vertex of~$F_i$ to~$R_i^j$ (step~9): in this case, an~existing
459 tree of~$F_i$ is extended, so the number of edges incident with it cannot decrease;\foot{%
460 This is the place where we needed to count the interior edges as well.}
461 \:all vertices have been processed (step~8): this can happen only in the final phase.
466 The Iterated Jarn\'\i{}k's algorithm finds the MST of the input graph in time
467 $\O(m\timesbeta(m,n))$, where $\beta(m,n):=\min\{ i: \log^{(i)}n \le m/n \}$.
470 Phases are finite and in every phase at least one edge is contracted, so the outer
471 loop is eventually terminated. The resulting subgraph~$T$ is equal to $\mst(G)$, because each $F_i$ is
472 a~subgraph of~$\mst(G_i)$ and the $F_i$'s are glued together according to the Contraction
473 lemma (\ref{contlemma}).
475 Let us bound the sizes of the graphs processed in the individual phases. As the vertices
476 of~$G_{i+1}$ correspond to the components of~$F_i$, by the previous lemma $n_{i+1}\le
477 2m_i/t_i$. Then $t_{i+1} = 2^{\lceil 2m/n_{i+1} \rceil} \ge 2^{2m/n_{i+1}} \ge 2^{2m/(2m_i/t_i)} = 2^{(m/m_i)\cdot t_i} \ge 2^{t_i}$,
480 \left. \vcenter{\hbox{$\displaystyle t_i \ge 2^{2^{\scriptstyle 2^{\scriptstyle\rddots^{\scriptstyle m/n}}}} $}}\;\right\}
481 \,\hbox{a~tower of~$i$ exponentials.}
483 As soon as~$t_i\ge n$, the $i$-th phase must be final, because at that time
484 there is enough space in the heap to process the whole graph. So~there are
485 at most~$\beta(m,n)$ phases and we already know (Lemma~\ref{ijphase}) that each
486 phase runs in linear time.
490 The Iterated Jarn\'\i{}k's algorithm runs in time $\O(m\log^* n)$.
493 $\beta(m,n) \le \beta(1,n) = \log^* n$.
497 When we use the Iterated Jarn\'\i{}k's algorithm on graphs with edge density
498 at least~$\log^{(k)} n$ for some $k\in{\bb N}^+$, it runs in time~$\O(km)$.
501 If $m/n \ge \log^{(k)} n$, then $\beta(m,n)\le k$.
505 Gabow et al.~\cite{gabow:mst} have shown how to speed this algorithm up to~$\O(m\log\beta(m,n))$.
506 They split the adjacency lists of the vertices to small buckets, keep each bucket
507 sorted and consider only the lightest edge in each bucket until it is removed.
508 The mechanics of the algorithm is complex and there is a~lot of technical details
509 which need careful handling, so we omit the description of this algorithm.
511 \FIXME{Reference to Chazelle.}
513 \FIXME{Reference to Q-Heaps.}
515 %--------------------------------------------------------------------------------
517 %\section{Verification of minimality}