Speeding Up Enumeration Algorithms with Amortized Analysis Takeaki Uno (National Institute of Informatics, JAPAN)

Motivation & Goal There have been proposed huge number of (output) polynomial time listing, enumeration, and generation algorithms <= Good construction schemes: Back tracking, Gray code, Binary partition, and Reverse search But, there are relatively few studies on Speeding up those algorithms (reducing complexities)

Motivation & Goal (2) Recently, enumeration algorithms are used in practical problems: data mining, bioinformatics, computational linguistics  Practical & theoretical fast algorithms are required However, reducing time complexity ( per output ) is not easy

Motivation & Goal (3) Reducing # of iterations  usually, hard ( or impossible if O(#output) ) Reducing time complexity of iterations  usually, difficult ( O(n 5 )  O(n 4 ) is possible, but O(n)  O(log n), O(1) is hard )  Scheme using amortized analysis is required we propose a scheme with amortized analysis

Almost enumeration algorithms are based on recursive calls ( enumeration tree ) Iterations on lower levels take less computation time Lower levels have much iterations Observation & Idea Move computation time from upper levels to lower levels ( computation time will be amortized ) # large time short # small time long

The shape of enumeration tree depends on problems ( not like merge sort ) Difficult to decide (1) to which iteration, and (2) how much amount How to Move? Our rule is adaptive, (1) holds and receives (2) from parent to children, recursively (3) proportional to the computation time of children …? top-down

Bound by Amortized Analysis O(iter) : amortized computation time per iteration T(x) : computation time of iteration x T : maximum T(x) of bottom level iterations α >1 : a constant If α T(x) - α ≤ ∑ T(y) for any x y is a child of x  O(iter) = O ( T ) Condition (1) ^ ^

Condition (1) does not hold If several x do not hold Condition (1) : α T(x) - α ≤ ∑ T(y)  Some iterations receive much computation time from parents If x receives > α T(x) move computation time to all descendants of x x

Bound by Amortized Analysis (2) Condition (2') If ( # descendants of x ) = Ω ( T(x) ) ( or T(x) - α ≤ ∑ T(y) for any x ) y is a child of x If αT(y) + α ≥ T(x) for any x and its child y,  O(iter) = O ( T × log ( T(root) ) ) Condition (2) T(x) # T(x) ^

(case 1) a leaf iteration takes much computation time  T increases (case 2) a parent takes much computation time than all the children  Condition (1) (2') will be violated (case 3) a child takes few computation time than other children  Condition (2) will be violated Undesired Occasion ^

In each iteration, we do Trimming: remove unnecessary parts of input of each iteration  for avoiding ( case 1 & 2 ) Balancing: take balance of computation time of children  for avoiding ( case 3 ) Procedure to Avoid

Matchings Algorithm satisfying Condition (1) O(|V|)  O( 1 ) Matroid Bases Algorithm satisfying Condition (1) m :#elements O(mn)  O(1) ( oracle calls ) n :rank Directed spanning trees Trimming & balancing satisfying Condition (2') & (2) O( |V| 1/2 )  O( log 2 |V| ) Bipartite Perfect matchings Trimming & balancing satisfying Condition (2') & (2) O( |V| )  O( log|V| ) Results ( previous best  new )

Input: Graph G = ( V, E ) Output: All matchings of G Simple binary partition ( divide and conquer ) (1) Choose a max degree vertex v (2) For each e of E(v), Recursive call of G + (e) E(v): Edges incident to v G + (e) : Removal of e and edges adjacent to e  Satisfies Condition (1) At bottom level, T(x) = O(1)  T = O(1) Matchings  O(1) per matching ( = per iteration ) v E(v) e G + (e) ^

Input: Matroid M = ( E, I ) Output: All bases of M (1) Choose e 1 of E (2-a) If { e 1, e i } are circuits for e 2,…, e k Recursive calls of each M / e i and M ＼ { e 1,…, e k } (2-b) If { e 1, e i } are cuts for e 2,…, e k Recursive calls of M / { e 1,…, e k } and each M ＼ e i (2-c) Otherwise Recursive calls of M / e 1 and M ＼ e 1  Satisfies Condition (1) At bottom level, T(x) = O(1) oracle calls Matroid Bases  O(1) oracle calls per base ( = per iteration ) e1e1 e2e2 e3e3 e4e4 e 1,…, e k e1e1

Input: Directed Graph G = ( V, A ), a vertex r Output: All directed spanning trees of G rooted at r Binary partition with Trimming and Balancing (1) Choose a vertex v (2) Partition E + (v) into E 1 and E 2 (3) Recursive calls inputting G ＼ E 1 and G ＼ E 2 E + (v): Out-going Arcs of v Directed Spanning Trees v E + (v) r

Trimming: (1) Remove arcs included in no directed spanning tree (2) Contract arcs included in all directed spanning trees Balancing: Choose v satisfying Condition (2)  Satisfies Condition (2), (2') At bottom level, T(x) = O(log |V|) ( = T ) Directed Spanning Trees (2)  O( log 2 |V| ) per tree ( = per iteration ) ^

Input: Bipartite Graph G = ( V, E ) Output: All perfect matchings of G Binary partition with Trimming and Balancing (1) Choose a vertex v (2) Partition E(v) into E 1 and E 2 (3) Recursive calls inputting G ＼ E 1 and G ＼ E 2 Bipartite Perfect Matchings v E(v)

Trimming: Remove edges (a) included in no perfect matching (b) included in all perfect matchings (c) adjacent to edges of (b) Contract consecutive degree 2 vertices Balancing: Choose v satisfying Condition (2)  Satisfies Condition (2), (2') At bottom level, T(x) = O(1) (= T ) Bipartite Perfect Matchings (2)  O( log |V| ) per perfect matching ( = per iteration ) ^

Conclusion Proposed a new amortized analysis Proposed a new speeding up scheme ( trimming and balancing ) Proposed fast enumeration algorithms for mathings matroid bases directed spanning trees bipartite perfect mathings

Future Works Improve other algorithms possible? undirected s-t paths minimal cuts of planer graphs directed paths of planer graphs impossible? minimal cuts directed s-t paths stable sets