CSE 548: Analysis of Algorithms Lectures 14 & 15 ( Dijkstra’s SSSP & Fibonacci Heaps ) Rezaul A. Chowdhury Department of Computer Science SUNY Stony Brook Fall 2012 Fibonacci Heaps ( Fredman & Tarjan, 1984 ) A Fibonacci heap can be viewed as an extension of Binomial heaps which supports DECREASE -KEY and DELETE operations efficiently. Binary Heap Binomial Heap Heap Operation ( worst-case ) ( amortized ) MAKE -HEAP Θ Θ 1 1 INSERT Ο Θ log ͢ 1 MINIMUM Θ Θ 1 1 EXTRACT -MIN Ο Ο log ͢ log ͢ UNION Θ Θ ͢ 1 DECREASE -KEY Ο log ͢ Ǝ DELETE Ο log ͢ Ǝ Fibonacci Heaps ( Fredman & Tarjan, 1984 ) A Fibonacci heap can be viewed as an extension of Binomial heaps which supports DECREASE -KEY and DELETE operations efficiently. Binary Heap Binomial Heap Heap Operation ( worst-case ) ( amortized ) MAKE -HEAP Θ Θ 1 1 INSERT Ο Θ log ͢ 1 MINIMUM Θ Θ 1 1 EXTRACT -MIN Ο Ο log ͢ log ͢ UNION Θ Θ ͢ Ο 1 DECREASE -KEY Ο ( worstlog case ͢ ) log ͢ Ο DELETE Ο ( worstlog case͢ ) log ͢ Fibonacci Heaps ( Fredman & Tarjan, 1984 ) A Fibonacci heap can be viewed as an extension of Binomial heaps which supports DECREASE -KEY and DELETE operations efficiently. Binary Heap Binomial Heap Fibonacci Heap Heap Operation ( worst-case ) ( amortized ) ( amortized ) MAKE -HEAP Θ Θ Θ 1 1 1 INSERT Ο Θ Θ log ͢ 1 1 MINIMUM Θ Θ Θ 1 1 1 EXTRACT -MIN Ο Ο Ο log ͢ log ͢ log ͢ UNION Θ Θ Θ ͢ Ο 1 1 DECREASE -KEY Ο Θ ( worstlog case ͢ ) log ͢ 1 Ο DELETE Ο Ο ( worstlog case͢ ) log ͢ log ͢ Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ 3. ͧ. ͘ ← 0 { empty min-heap } ͂ ← & 4. for each do INSERT ( ) 5. while ͪ ∈do ́ ͐ ͂, ͪ ͂ ƕ ∅ 6. EXTRACT -MIN ( ) 7. forͩ ←each ͂ do ͪ ∈ ̻͘͞ ͩ 8. if then ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 9. DECREASE -KEY ( ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 10. ͪ. ͘ ← ͩ. ͘ ƍ ͫ0,1 Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) Let and ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do ͢ Ɣ ́ ͐ ͡ Ɣ ́ ̿ 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ # INSERTS ͧ. ͘ ← 0 3. { empty min-heap } # EXTRACT -MINS ͂ ← & Ɣ ͢ 4. for each do INSERT ( ) # DECREASE -KEYS 5. while ͪ ∈do ́ ͐ ͂, ͪ Ɣ ͢ ͂ ƕ ∅ 6. EXTRACT -MIN ( ) ƙ ͡ 7. forͩ ←each ͂ do Total cost ͪ ∈ ̻͘͞ ͩ 8. if then ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 9. DECREASE -KEY ( ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 ). -/ 3/-/ͯ$) 10. ƙ ͢ ͗ͣͧͨ ƍ ͗ͣͧͨ ͪ. ͘ ← ͩ. ͘ ƍ ͫ0,1 ƍ ͡ ͗ͣͧͨ - . ͯ 4 Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) Let and ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do ͢ Ɣ ́ ͐ ͡ Ɣ ́ ̿ 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ For Binary Heap ( worst-case costs ): 3. ͧ. ͘ ← 0 { empty min-heap } ͂ ← & 4. for each do INSERT ( ) Ο 5. while ͪ ∈do ́ ͐ ͂, ͪ ͗ͣͧͨ ). -/ Ɣ logΟ ͢ ͂ ƕ ∅ 6. EXTRACT -MIN ( ) ͗ͣͧͨ3/-/ͯ$) Ɣ Οlog ͢ 7. forͩ ←each ͂ do ͪ ∈ ̻͘͞ ͩ - . ͯ 4 8. if then ͗ͣͧͨ Ɣ log ͢ ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 9. DECREASE -KEY ( ) Total cost ( worst-case ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 10. ∴ Ο ͪ. ͘ ← ͩ. ͘ ƍ ͫ0,1 Ɣ ͡ ƍ ͢ log ͢ Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) Let and ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do ͢ Ɣ ́ ͐ ͡ Ɣ ́ ̿ 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ For Binomial Heap ( amortized costs ): 3. ͧ. ͘ ← 0 { empty min-heap } ͂ ← & 4. for each do INSERT ( ) Ο 5. while ͪ ∈do ́ ͐ ͂, ͪ ͗ͣͧͨ ). -/ Ɣ 1 Ο ͂ ƕ ∅ 6. EXTRACT -MIN ( ) ͗ͣͧͨ3/-/ͯ$) Ɣ Οlog ͢ 7. forͩ ←each ͂ do ͪ ∈ ̻͘͞ ͩ - . ͯ 4 ( worst-case ) 8. if then ͗ͣͧͨ Ɣ log ͢ ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 9. DECREASE -KEY ( ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 10. Total cost ( worst-case ) ͪ. ͘ ← ͩ. ͘ ƍ ͫ0,1 ∴ Ο Ɣ ͡ ƍ ͢ log ͢ Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) Let and ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do Total͢ cost Ɣ ́ ͐ ͡ Ɣ ́ ̿ 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ 3. ͧ. ͘ ← 0 { empty min-heap } ͂ ← & 4. for each do INSERT ( ) ƙ ͢ ͗ͣͧͨ ). -/ ƍ ͗ͣͧͨ3/-/ͯ$) ͪ ∈ ́ ͐ ͂, ͪ 5. while do - . ͯ 4 ͂ ƕ ∅ ƍ ͡ ͗ͣͧͨ 6. EXTRACT -MIN ( ) Observation: 7. forͩ ←each ͂ do Obtaining a worst-case bound for a ͪ ∈ ̻͘͞ ͩ 8. if then sequence of INSERTS , EXTRACT -MINS ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 9. DECREASE -KEY ( ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 and DECREASE͢ -KEYS is ͢enough. 10. ͪ. ͘ ← ͩ. ͘ ƍ ͫ0,1 Amortized͡ bound per operation is ∴sufficient. Dijkstra’s SSSP Algorithm with a Min-Heap ( SSSP: Single-Source Shortest Paths ) Input: Weighted graph with vertex set and edge set , a weight function , and ́a source Ɣ ͐, ̿ vertex . ͐ ̿ Output: For all ͫ , is set to theͧ ∈ shortest ́ ͐ distance from to . ͪ ∈ ́ ͐ ͪ. ͘ ͧ ͪ Dijkstra-SSSP ( ) Let and ́ Ɣ ͐, ̿ , ͫ, ͧ 1. for each do Total͢ cost Ɣ ́ ͐ ͡ Ɣ ́ ̿ 2. ͪ ∈ ́ ͐ ͪ. ͘ ← ∞ 3. ͧ. ͘ ← 0 { empty min-heap } ͂ ← & 4. for each do INSERT ( ) ƙ ͢ ͗ͣͧͨ ). -/ ƍ ͗ͣͧͨ3/-/ͯ$) ͪ ∈ ́ ͐ ͂, ͪ 5. while do - . ͯ 4 ͂ ƕ ∅ ƍ ͡ ͗ͣͧͨ 6. EXTRACT -MIN ( ) Observation: 7. forͩ ←each ͂ do For ͪ ∈ ̻͘͞ ͩ 8. if then the best possible ). -/ bound 3/-/ͯ$)is Θ . ͪ. ͘ Ƙ ͩ. ͘ ƍ ͫ0,1 ͢ ͗ͣͧͨ ƍ ͗ͣͧͨ 9. DECREASE -KEY ( ) ͂, ͪ, ͩ. ͘ ƍ ͫ0,1 ( else violates sorting lower bound͢ log ) ͢ 10. 0,1 ͪ. ͘ ← ͩ. ͘ ƍ ͫ Perhaps can be improved͡ to͗ͣͧ ͨ - . .ͯ 4 o ͡ log ͢ Fibonacci Heaps from Binomial Heaps A Fibonacci heap can be viewed as an extension of Binomial heaps which supports DECREASE -KEY and DELETE operations efficiently. But the trees in a Fibonacci heap are no longer binomial trees as we will be cutting subtrees out of them. However, all operations ( except DECREASE -KEY and DELETE ) are still performed in the same way as in binomial heaps. The rank of a tree is still defined as the number of children of the root, and we still link two trees if they have the same rank. Implementing DECREASE -KEY ʚ ¤, Î, Á ʛ DECREASE -KEY ( ): One possible approach is to cut out the subtree rooted¤, at Î, Áfrom , reduce the value of to , and insert that subtree into the rootͬ list of͂ . ͬ ͟ ͂ Problem: If we cut out a lot of subtrees from a tree its size will no longer be exponential in its rank. Since our analysis of EXTRACT -MIN in binomial heaps was highly dependent on this exponential relationship, that analysis will no longer hold. Solution: Limit #cuts among the children of any node to 2. We will show that the size of each tree will still remain exponential in its rank. When a 2nd child is cut from a node , we also cut from its parent leading to a possible sequence of cutsͬ moving up toͬwards the root. Analysis of Fibonacci Heap Operations Recurrence for Fibonacci numbers : 0 ͚͝ ͢ Ɣ 0, ͚) Ɣ ʰ 1 ͚͝ ͢ Ɣ 1, ͚)ͯͥ ƍ ͚)ͯͦ ͙͙ͣͨͦͫͧ͜͝. We showed in a pervious lecture: , ͥ ) ) ͚) Ɣ ͩ & Ǝ&Ȩ where and are the roots . ͥͮ ͩ ͥͮ ͩ ͦ &Ɣ ͦ &Ȩ Ɣ ͦ ͮ Ǝ ͮ Ǝ 1 Ɣ 0 Analysis of Fibonacci Heap Operations Lemma 1: For all integers , .