Shoudai, Takayoshi

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Title The Complexity of Topological Sorting AlgorithmsAuthor(s)Shoudai, TakayoshiCitation数理解析研究所講究録 (1989), 695: 169-177Issue Date1989-06URL http://hdl.handle.net/2433/101392RightType Departmental Bulletin PaperTextversionpublisherKyoto University169Topological Sorting NLOG-The Complexity of Topological Sorting Algorithms-Takayoshi ShoudaiDepartment of Mathematics, Kyushu UniversityWe consider the following problem: Given a directed acyclic graph $G$ and vertices$s$ and $t$ , is $s$ ordered before $t$ in the topological order generated by a given topologicalsorting algorithm? For known algorithms, we show that these problems are log-spacecomplete for NLOG. It also contains the lexicographically first topological sortingproblem. The algorithms use the result that NLOG is closed under conplementation.1. IntroductionThe topological sorting problem is, given a directed acyclic graph $G=(V, E)$ , to finda total ordering of its vertices such that if $(v, w)$ is an edge then $v$ is ordered before $w$ .It has important applications for analyzing programs and arranging the words in theglossary [6]. Moreover, it is used in designing many efficient sequential algorithms,for example, the maximum flow problem [11].Some techniques for computing topological orders have been developed. Thealgorithm by Knuth [6] that computes the lexicographically first topological orderruns in time $O(|E|)$ . Tarjan [11] also devised an $O(|E|)$ time algorithm by employ-ing the depth-first search method. Dekel, Nassimi and Sahni [4] showed a parallelalgorithm using the parallel matrix multiplication technique. Ruzzo also devised asimple $NL^{*}$-algorithm as is stated in [3]. Hence this problem is in $NC^{2}$ . However,any completeness result does not seem to be known as to the exact complexity of thetopological sorting problem.695 1989 169-177In this paper, we consider the decision problems for the orders generated by topo-logical sorting algorithms. We show that the problems for the above four algorithmsare all NLOG-complete. But it is not known whether all possible topological sortingproblems are NLOG-complete. Immermann [5] and Szelepcs\’enyi [10] showed thatNLOG is closed under complementation. By using this result, we can give NLOG-alogrithms computing the Iexicographically first toplogical sort and other problems.It should be noted that the “lexicographically first” property often makes some problems P-complete [7,8,9]. But the topological sorting with this property remains inNLOG.2. Topological sortWe classify the known topological sorting algorithms into the following four types.Let $G=(V, E)$ be a directed acyclic graph, where $V=\{1, \ldots, n\}$ and $|E|=m$ .(K) Knuth [6]: For each vertex $v\in V$ , we assume an adjacency list $adj[v]$ of ver-tices linked by an edge from $v$ and an integer $count[v]$ for the number of predecessorsof $v$ . The vertices whose count has been reduced to zero but which have not yet beenordered are stored in a heap [11]. Two operations insert and deletemin are possibleon a heap. The algorithm is as follows. Initially the heap $h$ is empty.beginfor $v:=1$ to $n$ doif $count[v]=0$ then insert(v, $h$);for $i:=1$ to $n$ dobegin$v$ :=deletemin(h); $order[i]:=v$for each $u\in adj[v]$ dobegin$count[u]:=count[u]-1$ ;if $count[u]=0$ then insert(u, $h$);2endendend.Since deletemin and insert take $O(\log n)[11]$ and each operation executes $n$times, this algorithm has an $O(n\log n+m)$ time bound. This algorithm generatesthe lexicographically first topological order on $G$ . We call this order $l$ ft-order forshort.(T) Tarjan [11]: We assume an adjacency list which is ordered lexicographcally.The depth-first serch is executed to make a forest. Then we order the vertices indecreasing order as they are unstacked. This algorithm runs in $O(n+m)$ time.(D) Dekel, Nassimi and Sahni [4]: We assume an adjacency matrix. By usingmatrix multiplication, we can compute the length of the longest path from any vertexwith no predecessors to each vertex. Then vertices are sorted in nondecreasing orderof their lengths. This is an $NC^{2}$-algorithm.(R) Ruzzo (Cook [3]): We compute the transitive closure of an adjacency matrix.This gives the number of predecessors of each vertex by summing the columns. Thenwe sort vertices by these numbers.Generally the problem $TS(A)$ is defined as follows: Given a directed acyclic graph$G=(V, E)$ and two distinguished vertices $s$ and $t$ , determine whether $s$ ordered before$t$ in the order generated by a given topological sorting algorithm $A$ . We define thelexicographically first topological sorting problem (LFTS) as TS(K).3. Topological sortings are NLOG-hardWe will show that the problem LFTS is NLOG-hard. The topological sorting prob-lems for the other algorithms mentioned in Section 2 can also be shown NLOG-hardin a similar way.We identify the vertices with its numbers. For $u,$ $v\in V,$ $uarrow^{*}v$ denote theexistence of a path from $u$ to $v$ , and $uarrow+v$ denote the fact $u\neq v$ and $uarrow^{*}v$ . In3172addition, $u\prec v$ denote the fact that $u$ is ordered before $v$ in the lft-order. For the$l$ ft-order of any directed acyclic graph, the following lemma holds.Lemma 3.1. For any $u,$ $v\in V,$ $u\prec v$ if and only if either $u\sim^{+}v$ or there exists avertex $x$ such that $u\prec x_{f}xarrow^{*}v$ and $x>u$ .Proof. If $u\prec v$ and $u\star^{+}v$ , let $x$ be the rightmost vertex in the $order\prec such$ that$u\prec x$ and $xarrow*v$ . Its vertex may be $v$ . If $x<u$ , it contradicts the fact that $\prec is$the lexicographically first order. Thus $x>u$ . Conversely, it is easy to see $u\prec v$ ifthe condition holds. $\square$Corollary 3.2. Suppose $u$ is the largest vertex in V. Then $u\prec v$ if and only if$uarrow^{+}v$ .Proof. Straightforward by Lemma 3.1. $\square$Theorem 3.3. $TS(A)$ is NLOG-hard for $A=K,$ $T,$ $D$ , R.Proof. The monotone graph accessibility problem (MGAP) is described as follows:Given a directed acyclic graph $G$ and vertices $s$ and $t$ , then $\det^{\backslash }ern\dot{u}ne$ whether $t$is reachable fr$oms$ . It is known that MGAP is NLOG-complete. We first give areduction from MGAP to LFTS $(TS(K))$ . For an instance $(G, s, t)$ of MGAP, werenumber $t$he vertices so that $s$ has the largest number of $V$ . If there exists a pathfrom $s$ to $t,$ $s$ must be ordered before $t$ in any topological order. Conversely if $s$ isordered before $t$ in the $l$ ft-order, Corollary 3.2 implies that there exists a path from$s$ to $t$ . It is clear that this reduction is computable in log-space.Reductions from MGAP to the other problems are $si_{1}nilar$ . For TS(T), we renum-ber vertices so that $sh$as the smallest number. Then $s$ is the root of the first tree inthe depth-first forest. If $t$ is reachable from $s$ , this tree contains $t$ . Otherwise, $t$ is inone of the other trees. For TS(D) and TS(R), we attach a new directed n-chain byidentifying the sink of the n-chain with the vertex $s$ where $n=|V|$ . Then $s$ has thelarger key than that of any vertex which is not reachable from $s$ . $\square$417 $d$4. NLOG-algorithms for the TS problemsIn this section, we will present an NLOG-algorithm for LFTS $(TS(K))$ and describeNLOG-algorithms for $TS(A)(A=T, D, R)$ . The class NLOG is closed under com-plementation $[5,10]$ . Hence NLOG contains the complement of MGAP $(cc\succ MGAP)$ .We use the MGAP and its complement to see reachabilities between vertices.For $u\in V,$ $R(u)$ denote the se$t\{v\in V|varrow^{*}u\}$ . We consider a sequence$u_{1},$ $u_{2},$ $\cdots,$ $u_{p}$ of vertices defined as follows:$u_{1}$ $:=$ $\max R(s)\cap R(t)$ .$u_{i}$ $:=$ $\max\{v\in R(s)\cap R(t)|u_{i-1}arrow^{+}v\}(i=2,3, --)$ .We assume that the set function $\max$ returns zero if the argument is empty. Let$u_{p}$ be the last vertex whose value is positive.Lemma 4.1. For any $u\in R(s)\cap R(t),$ $u_{i}\prec u$ if and only if $u_{i}arrow^{+}u(i=1,2, \cdots,p)$ .Proof. The proof is by induction on $i$ . Suppose $u_{1}\prec u$ for $u\in R(s)\cap R(t)$ . If$u_{1}\star^{+}u$ , there exists a vertex $x$ in $R(s)\cap R(t)$ which is larger than $u_{1}$ by Lemma3.1. This contradicts the definition of $u_{1}$ . Thus $u_{1}arrow^{+}u$ . Suppose $u_{i}\prec u$ . If $u_{i}\star^{+}u$ ,there $e$xists a vertex $x$ such that $u_{i}\prec x,$ $xarrow^{*}u$ and $x>u_{i}$ . So $x\in R(s)\cap R(t)$ .Therefore $u_{i-1}\prec u_{i}$ and $u_{i}\prec x$ imply $u_{i-1}\prec x$ . By the induction hypothesis,$u_{i-1}arrow+x$ . This contradicts the definition of $u..\square$Corollary 4.2. If $u_{p}\prec w$ , then $w\not\in R(s)\cap R(t)$ .Proof. Straightforward by Lemma 4.1. $\square$Lemna 4.3. We assume that th$eree$xists $i(1\leq i\leq p)$ such that $w’<u_{i}$ for any$w’\in R(s)-R(t)$ . Then for any $w\in R(s)-R(t)_{f}u_{t}\prec w$ if and only if $u$. $arrow^{+}w$ .Proof. If $u_{i}\prec w$ and $u_{i}\star^{+}w$ , then there exists $x$ which satisfies $u_{i}\prec x$ and $x>u.$ .Thus, by the assumption, $x\in R(s)\cap R(t)$ . This contradicts the definition of $u_{k}$ since$u_{i}arrow^{+}x$ by Corollary 4.2. $\square$5$\perp lq$Lemma 4.4. For $w\in R(s)-R(t)$ , if either $w>u_{1}$ or there exists $i(2\leq i\leq p)$such that $u_{i-1}\prec w$ and $w>u_{i}$ , then $u_{p}\prec w$ .Proof. Suppose $w\prec u_{p}$ . Since $w\star^{+}u_{p}$ , there exists $a$ $ve$rtex $x$ which is in$R(s)\cap R(t)$ . This fact and $u_{i-1}\prec w$ (if $i\geq 2$) imply $u_{i}\prec x$ . This contradicts thedefinition of $u_{i}$ . $\square$From now on, we assume that there does not exist a path between $s$ and $t$ . Let$v_{s}= \max\{v\in R(s)-R(t)|u_{p}\prec v\}$ and $v_{t}= \max\{v\in R(t)-R(s)|u_{p}\prec v\}$ . (If$R(s)\cap R(t)=\phi$ , let $v_{s}= \max R(s)$ and $v_{t}= \max R(t).)$Lemma 4.5. $v_{s}<v_{t}$ if and only if $s\prec t$ .Proof. Suppose $v_{t}\prec s$ . By Lemma 3.1, there exists a vertex $x$ such that $v_{t}\prec x$ ,$xarrow*s$ and $x>v_{t}$ . Since $u_{p}\prec v_{t},$ $u_{p}\prec x$ . Thus $x\in R(s)-R(t)$ by Corollary 4.2and $xarrow^{*}s$ . It contradicts the fact $v_{s}<v_{t}$ . So $s\prec v_{t}$ , thus, $s\prec t$ . Conversely, if$s\prec t$ , then $v_{s}\prec t$ . By Lemma 3.1, there exists a vertex which is 1$ar$ger than $v_{s}$ . Thus$v_{s}<v_{t}$ . $\square$When we can verify that there exists $i(2\leq i\leq p)$ such $th$at $u_{i-1}>w$ for any$w\in R(s)-R(t)$ , if $u:-1arrow^{+}w$ and $u_{i}<w$ , then $u_{p}\prec w$ by Lemma 4.3 and Lemma4.4. Therefore, under this assumption, $v_{s}$ equals to $\max\{v\in R(s)-R(t)|u_{i-1}arrow^{+}v\}$if this value is larger than $u_{i}$ .Theorem 4.4. LFTS is NLOG-complete.Proof. The following algorithm can decide in NLOG whether $s\prec t$ . We can decidethe reachability and unreachability by using the MGAP and co-MGAP.beginif $sarrow^{+}t$ then acceptelse if $tarrow^{+}s$ then reject;$v_{s}$ $:=naxR(s)-R(t);v_{t}$ $:= \max R(t)-R(s);u:=\max R(s)\cap R(t)$ ;61 $/i$)while true dobeginif $u<v_{s}$ or $u<v_{t}$ thenif $v_{s}<v_{t}$ then accept else reject;$v_{s}$ $:= \max\{v\in R(s)-R(t)|uarrow^{+}v\}$ ;$v_{t}$ $:= \max\{v\in R(t)-R(s)|uarrow^{+}v\}$ ;$u$ $:=nax\{v\in R(s)\cap R(t)|uarrow^{+}v\}$ ;endend.The variable $u$ is stored the value $u_{i}(i=1,2, \cdots)$ . It is easy to see that thisalgorithm correctly decide the order between $s$ and $t$ by Lemma 4.3, Lemma 4.4 andLemma 4.5. The reachability between vertices is computed in NLOG by enumeratingall vertices and employing the MGAP and co-MGAP. $\square$TS(R) is dso NLOG-complete since the number of predecessors can be computedin nondeterministic log-space by using Reach.For computing the length of the longest path, we use the following problem:Given a directed acyclic graph $G$ , vertices $s$ and $t$ and a nonnegative integer $l$ , decidewhether there exists $a$ path whose lengt $h$ is longer then $l$ . This problem is solvedin NLOG. By employing it and its complement, we can construct the function forcomputing the length of the longest path. This implies an NLOG-algorithm forTS(D).TS(T) uses the depth-first search method, but this seaching is not known to bein NLOG [1]. However, we can solve TS(T) in NLOG for $a$ directed acyclic graph.For any vertex $v$ , the root of the tree which contains $v$ is the smallest vertex suchthat $v$ is reachable from it. This ro$ot$ can be found in NLOG by using Reach. Wecompute the roots for $s$ and $t$ , respectively. Then the root of $s$ is larger than that of$t$ if and only if $s$ is ordered before $t$ . If they are identical, we find the new root for7$\perp lO$$s$ (resp. t) such that it is one of the children of the current root and the subtree atthe new root contains $s$ (resp. $t$ ). Then we compare them. These roots also can befound in NLOG. We iterate this process until they are not identical.5. Concluding RemarksCook [3] illustrates a taxonomy of problems in parallel computations. The classNLOG locates between $NC^{1}$ and $NC^{2}$ . The sorting problem is solvable by circuitsof depth $O(\log n)$ with polynomi$a1$ gates [2]. On the other hand, we have shown thatthe known topological sorting $pr$oblems are all NLOG-complete. Since it seems thatNLOG differs from $NC^{1}$ , there nay exist no $NC^{1}$-algorithms computing one of thefour types topological sorting problems. But it is an open question whethe $r$ there isan $NC^{1}$ topologic$a1$ sorting algorithm.Acknowledgement. The author would like to thank Prof. S. Miyano for manyhelpful discussions and much thoughtful criticism.$\ovalbox{\tt\small REJECT}\yen X\mathbb{B}$[1] A. Aggarwal and R.J. Anderson, A random NC algorithm for depth first search,Proc. 19th ACM Symp. Theory of Comput. (1987) 325-334.[2] M. Ajtai, J. Koml\’os and E. Szemer\’edi, An $O(n\log n)$ sorting network, Proc.15th ACM Symp. Theory of Comput. (1983) 1-9.[3] S.A. Cook, A taxonomy of problems with fast parallel algorithms, Inform. Contr.64 (1985) 2-22.[4] E. Dekel, D. Nassimi and S. Sahni, Parallel matrix and graph algorithms, SIAMJ. Comput. 10 (4) (1981) 657-675.[5] N. Immermann, Nondeterministic space is closed under complement, TechnicalReport, Department of Computer Sicence, Yale University, July (1987).8[6] D. Knuth, (($The$ Art of Computer Programming,” Vol.1, Addison-Wesley, Read-ing, Mass. (1968).[7] S. Miyano, The lexicographically first maximal subgraph problems: P-completeness and NC algorithms, Proc. 13th ICALP (Lecture Notes in ComuterScience 267) (1987) 425-434.[8] S. Miyano. A parallelizable lexicographicaly first maximal edge-induced sub-graph problem, $Info_{\wedge}rm$ , Process. Lett. 27 (1988) 75-78.[9] J.H. Reif, Depth-first search is inherently sequential, Inform. Process. Lett. 20(1985) 229-234.[10] R. Szelepcs\’enyi, The method of forcing for nondeterministic automata, Bulletinof the EATCS 33 (1987) 96-100.[11] R.E. Tarjan, “Data Structures and Network Algorithms,” CBMS-NSF Region$a1$Conference Series in Applie$d$ Mathematics 44, SIAM, Philadelphia, Pennsylva-nia (1983).9

Topological Sorting の NLOG 完全性について(計算アルゴリズムと計算量の基礎理論)

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Topological Sorting の NLOG 完全性について(計算アルゴリズムと計算量の基礎理論)

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