Design and Analysis of Algorithms - PowerPoint PPT Presentation

�������� Design and Analysis of Algorithms ����������������������� ������ �����ก��� ������ ��� ��������� ���������������������� �������������������� ��. ��. ����� ��������������ก�� ������������������������������ก������� ������ก�� �����������ก�������������� ��������������� ������������กก�������ก��������� ������ก�������������� 2542 http://www.cp.eng.chula.ac.th/faculty/spj Outline Advanced Data Structures � Lazy Union + Cascading Cut � Decrease Key � Analysis Fibonacci Heaps http://www.cp.eng.chula.ac.th/faculty/spj

Lazy Bino. Heap vs. Fibonacci Fibonacci Heaps • Make-Heap : π ( 1 ) : π ( 1 ) � Lazy union : π (1) • Insert : π ( 1 ) : π ( 1 ) � Cascading cut in Decrease-Key • Minimum : π ( 1 ) : π ( 1 ) � achieve π (1) amortized time in Decrease-Key • Extract-Min : ν ( log n ) * : ν ( log n ) * � maintain O(log n ) amortized time in Extract-Min • Union : π ( 1 ) : π ( 1 ) • Decrease-Key : π ( log n ) : π ( 1 ) * • Delete : π ( log n ) : O( log n ) * * amortized cost http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Fibonacci Heaps : Simple Cut Fibonacci Heaps : Simple Cut H H 20 2 5 20 2 5 21 9 32 20 21 9 32 20 11 13 40 4 13 40 18 18 x Fib-Heap-Decrease-Key( H, x, 4 ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Fibonacci Heaps : Simple Cut Simple Cut Doesn�t Work H H 20 4 2 5 5 18 21 9 32 20 9 32 20 13 40 11 13 40 x 18 � Decrease-Key : Simple Cut is π (1). Fib-Heap-Delete( H, x ) � More than one tree of the same rank. { � Trees are not necessarily binomial trees Fib-Heap-Decrease-Key( H, x, -inf ) Fib-Heap-Extract-Min( H ) � Extract-Min is no longer amortized O( log n ). } http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Simple Cut Dosen�t Work Simple Cut Dosen�t Work H H Λ⊗ 5 5 9 32 20 9 32 20 11 13 40 11 13 40 Fib-Heap-Delete( H, x ) Fib-Heap-Delete( H, x ) { { Fib-Heap-Decrease-Key( H, x, -inf ) Fib-Heap-Decrease-Key( H, x, -inf ) Fib-Heap-Extract-Min( H ) Fib-Heap-Extract-Min( H ) } } http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Simple Cut Dosen�t Work Lazy Binomial Heap : Analysis H � Invarient : one baht credited at each root node The rank of the root is 5 no longer O( log n ) for future consolidating task. 9 32 20 H 11 13 40 Fib-Heap-Delete( H, x ) { Fib-Heap-Decrease-Key( H, x, -inf ) Fib-Heap-Extract-Min( H ) } http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Lazy Binomial Heap : Analysis Lazy Binomial Heap : Analysis � Insert : amortized cost = 2 � Extract-Min : amortized cost = O( log n ) H H � one for list insertion + one stored at the root http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Lazy Binomial Heap : Analysis Lazy Binomial Heap : Analysis � Extract-Min : amortized cost = O( log n ) � Extract-Min : amortized cost = O( log n ) O( log n ) children H consolidate Delete minimum root using the credit stored at the root http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Simple Cut Doesn�t Work Cascading Cut rank is π ( n ) Extract-Min � Guarantees any node rank to be O( log n ) � Extract-Min : O( log n ) delete root Amortized cost � Decrease-Key : π ( 1 ) consolidate costs π ( n ) To maintain O( log n ) Extract-Min, node rank must be O( log n ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Marking Nodes Cascading Cuts H � Mark a (nonroot) node the first time that it loses 20 2 4 a child (because of a simple cut). 12 21 5 19 31 12 � If a marked node loses another child, then cut it 30 22 33 45 9 32 20 from its parent causing cascading cut 7 11 13 40 28 18 Fib-Heap-Decrease-Key( H, x, 7 ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Cascading Cuts Cascading Cuts H H 2 8 2 20 7 4 20 7 4 18 12 21 5 18 12 21 5 19 31 12 19 31 12 30 30 22 33 45 22 33 45 9 32 20 9 32 8 11 13 40 28 13 40 28 Fib-Heap-Decrease-Key( H, x, 8 ) Fib-Heap-Decrease-Key( H, x, 11 ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Cascading Cuts Cascading Cuts H H 8 11 2 8 11 9 2 20 7 4 20 7 4 12 21 12 21 18 5 18 5 19 31 12 19 31 12 30 30 22 33 45 22 33 45 9 32 32 40 28 40 28 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Cascading Cuts Cascading Cuts : Two Issues H � Why is Decrease-Key Θ (1) ? 8 11 9 5 2 20 7 4 12 21 � Why is any node rank O( log n ) ? 18 32 19 31 12 40 30 22 33 45 � i.e., Extract-Min is O( log n ) 28 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Decrease-Key is Θ (1) Decrease-Key is Θ (1) � Let w be a cut node � Let w be a cut node � The amortized cost of Decrease-Key is 4 � The amortized cost of Decrease-Key is 4 1 1 1 w ... � two is credited at w �s parent node � two is credited at w �s parent node A 2 2 � one is for a simple cut at node w � one is for a simple cut at node w w A � one is credited at w after being a new root One baht at each root node will be used during consolidation. http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Decrease-Key is Θ (1) Decrease-Key is Θ (1) x 1 0 1 A 0 0 z z 2 0 2 +2 x 0 0 y 0 0 y 0 A B C B C Cut( x ) Cut( x ) amortized cost : (2 amortized cost : (2+1 actual cost : actual cost : (1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Decrease-Key is Θ (1) Decrease-Key is Θ (1) +1 0 x 1 x 1 1 1 A A 0 0 z z 2 2 y 0 y 0 0 0 B C B C Cut( x ) Cut( y ) amortized cost : (2+1+1 amortized cost : (2+1+1)+ actual cost : (1 actual cost : (1)+ http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Decrease-Key is Θ (1) Decrease-Key is Θ (1) 1 x 1 x y 1 0 1 A A C 0 0 z z 4 2 4 +2 y 0 0 0 B C B Cut( y ) Cut( y ) amortized cost : (2+1+1)+(2+ amortized cost : (2+1+1)+(2+1 actual cost : (1)+ actual cost : (1)+(1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Decrease-Key is Θ (1) Decrease-Key is Θ (1) 1 x 1 x y y 1 0 1 1 1 A C A C +2 0 2 0 z z +1 4 4 2 0 0 B B Cut( y ) Cut( y ) -> Cut( z ) amortized cost : (2+1+1)+(2+1+1 amortized cost : (2+1+1)+(2+1+1) actual cost : (1)+(1 actual cost : (1)+(1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Decrease-Key is Θ (1) Money at Marked Nodes 1 x z y � Four bahts are used to cut node w whose parent 1 1 1 A C 2 is node v 0 � two bahts are transferred to v B � one baht is for the cut � one baht is stored on w for future consolidation Cut( y ) -> Cut( z ) v 2 amortized cost : (2+1+1)+(2+1+1) w actual cost : (1)+(1+1) 0 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Money at Marked Nodes Money at Marked Nodes � If node v loses only one child, it has two bahts � If node v loses two children, it has four bahts � two bahts are tranferred to its parent (may cause � one baht is for future cut (caused by cascading cut) further cut). � the other baht is for future consolidation � one baht is for cutting the node � another baht is for future consolidation v 1 1 1 1 2 v 0 2 2 v v 4 2 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Ranks of Children Ranks of Chlidren x x Rank of x is i ... v i -1 ... v 3 v 2 v 1 v i Rank of v 1 ≥ 0 Rank of v i -1 ≥ i -3 Rank of v 1 ≥ 0 Rank of v 2 ≥ 0 Rank of v i ≥ i -2 Rank of v 3 ≥ 1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Ranks of Chlidren Ranks of Chlidren x x Rank of v 1 ≥ 0 Rank of v 1 ≥ 0 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Ranks of Chlidren Ranks of Chlidren x x Rank of x ≥ i -1 Rank of x is i ... v i -1 ... v 3 v 2 v 1 v i v i Rank of v 1 ≥ 0 a node could have lost at most one child Rank of v i -1 ≥ i -3 Rank of v 2 ≥ 0 Rank of v i ≥ i -2 Rank of v i ≥ i -1 i -2 Rank of v 3 ≥ 1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj