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 � Priority Queue : ADT � Binary Heap � Binomial Trees Binomial Heaps � Binomial Heaps � Operations http://www.cp.eng.chula.ac.th/faculty/spj

Priority Queue : ADT Binary Heap 4 • Make-Heap() • Insert( H, x ) 10 30 π ( log n ) • Minimum( H ) 14 20 43 100 • Extract-Min( H ) 17 • Union( H 1 , H 2 ) 4 10 30 14 20 43 100 17 • Decrease-Key( H, x, k ) • Delete( H, x ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binary Heap : Insert Binary Heap : Insert 4 4 Bubble up 10 30 7 30 π ( log n ) 14 20 43 100 10 20 43 100 17 7 17 14 4 10 30 14 20 43 100 17 7 4 7 30 10 20 43 100 17 14 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binary Heap : Extract-Min Binary Heap : Extract-Min 4 4 7 30 7 30 10 20 43 100 10 20 43 100 17 14 17 14 4 7 30 10 20 43 100 17 14 4 7 30 10 20 43 100 17 14 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binary Heap : Extract-Min Binary Heap : Extract-Min 14 7 Percolate down 10 7 30 30 π ( log n ) 10 20 43 100 14 20 43 100 17 17 14 7 30 10 20 43 100 17 7 10 30 14 20 43 100 17 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binary Heap : Union Binary Heap : Union 7 4 7 10 50 10 30 23 30 14 20 43 100 14 20 43 100 17 17 4 50 23 7 10 30 14 20 43 100 17 4 50 23 7 10 30 14 20 43 100 17 4 50 23 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binary Heap : Union Binary Heap : Union 7 7 10 10 30 30 14 20 43 100 14 20 43 100 17 4 50 23 17 4 50 23 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binary Heap : Union Binary Heap : Union 7 7 10 10 30 30 14 20 43 100 4 20 43 100 17 4 50 23 17 14 50 23 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binary Heap : Union Binary Heap : Union 7 7 10 4 30 30 4 20 43 100 10 20 43 100 17 14 50 23 17 14 50 23 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binary Heap : Union Binary Heap : Union : Analysis n 4 x h 2 h 7 lg n 30 � � n n ∴ � � x 3 � O ( h ) O ( n ) 2 3 h � 2 � 10 20 43 100 h 0 ∴ n x 2 2 2 17 14 50 23 n x 1 2 1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binary Heap & Binomial Heap Binomial Trees B 0 B 1 B 2 B 3 B 4 • Make-Heap : π ( 1 ) : π ( 1 ) • Insert : π ( log n ) : ν ( log n ) • Minimum : π ( 1 ) : ν ( log n ) • Extract-Min : π ( log n ) : π ( log n ) B k • Union : ν ( n ) : ν ( log n ) • Decrease-Key : π ( log n ) : π ( log n ) • Delete : π ( log n ) : π ( log n ) B k -1 B k -1 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Properties of Binomial Trees Properties of Binomial Trees B 4 # nodes Root of B 4 has degree 4 � For the binomial tree B k 1 = 4 C 0 � there are 2 k nodes 4 = 4 C 1 Binomial 6 = 4 C 2 � the height of the tree is k Coef. 4 = 4 C 3 � there are exactly k C i nodes at depth i 1 = 4 C 4 � the root has degree k The height of B 4 is 4 2 4 � The max. degree of any node in an n -node binomial tree is lg n http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heaps Heap-Ordered Property � A list of binomial trees satisfying these properties 20 2 5 � each binomial tree is heap-ordered H � (the root of a tree contains the smallest key in the tree) 12 21 9 32 20 � there is at most one B k for any given k in the heap � (an n -node binomial heap consists of at most 1 + lg n 30 11 13 40 binomial trees) 18 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Structural Property Binomial Heaps: Minimum B 0 B 2 B 3 20 20 20 2 2 5 H 12 21 9 32 20 B 3 B 2 B 0 30 11 13 40 H = ( B 3 , B 2 , B 0 ) has 2 3 + 2 2 + 2 0 = 13 10 = 1101 2 nodes has 4 bits The binary representation of 13 10 18 The binary representation of n has (1 + lg n ) bits π ( #trees )= O( 1 + lg n ) = O( log n ) An n -node binomial heap has at most (1 + lg n ) binomial trees http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heap : Union Binomial Heap : Union 20 2 5 2 5 H 1 H 1 12 21 9 32 20 12 21 9 32 20 20 30 11 13 40 30 11 13 40 18 18 10 10 8 8 H 2 H 2 31 12 31 12 44 44 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binomial Heap : Union Binomial Heap : Union 10 2 5 2 5 H 1 H 1 20 12 21 9 32 20 12 21 9 32 20 B 2 30 11 13 40 30 11 13 40 18 18 10 8 8 20 H 2 B 1 H 2 31 12 31 12 B 2 44 44 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heap : Union Binomial Heap : Union 10 10 5 5 2 H 1 H 1 20 20 9 32 20 9 32 20 12 21 11 13 40 11 13 40 30 2 B 3 18 18 8 8 12 21 31 12 31 12 30 H 2 H 2 44 44 B 3 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binomial Heap : Union Binomial Heap : Union 5 10 10 H 1 H 1 20 20 9 32 20 11 13 40 2 2 18 5 8 8 12 21 12 21 9 32 20 31 12 31 12 30 30 H 2 H 2 11 13 40 44 44 18 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heap : Union Binomial Heap Union : Analysis ≤ 1+lg n 1 roots ≤ 1+lg n 2 roots 2 10 ... ... H 1 20 5 8 12 21 H 1 H 2 n 1 nodes n 2 nodes 9 32 20 31 12 30 Binomial_Heap_Union 11 13 40 44 ≤ 2+lg n 1 + lg n 2 < 2+ lg n + lg n = O( log n ) 18 ... H n = n 1 + n 2 nodes http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Binomial Heap Union : Analysis Binomial Heap Union : Analysis B 0 B 2 B 3 B 0 B 2 � T 1 : Union H 1 and H 2 of size n 1 and n 2 , respectively. � T 2 : Carry-ripple adding numbers n 1 and n 2 in binary H 1 H 2 � T 1 = O( T 2 ) 13 nodes 5 nodes B 0 B 2 B 3 Binomial_Heap_Union � Let n = n 1 + n 2 ⇒ 1 + lg n bits H 1 � T 2 = Θ ( 1 + lg n ) = Θ ( log n ) B 1 B 4 1 1 0 1 1 0 1 � T 1 = O( log n ) H 1 1 0 1 1 0 0 1 0 18 nodes http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heap : Insert Building Binomial Heap Binomial_Heap_Insert � Build an n -node binomial heap by n successive ... insertions H H * � An insertion costs O( log n ) Binomial_Heap_Union � n insertions cost O( n log n ) � Correct but not tight ! ... O( log n ) H http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Building Binomial Heap Building Binomial Heap H H 0 0 0 1 0 0 b 0 b 1 b 2 b 0 b 1 b 2 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Building Binomial Heap Building Binomial Heap H H 0 1 0 1 1 0 b 0 b 1 b 2 b 0 b 1 b 2 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Building Binomial Heap Building Binomial Heap H H 0 0 1 1 0 1 b 0 b 1 b 2 b 0 b 1 b 2 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Building Binomial Heap Building Binomial Heap H H 0 1 1 1 1 1 b 0 b 1 b 2 b 0 b 1 b 2 http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj

Building Binomial Heap Binomial Heap : Extract Min. � Calling binary counter Increment s n times cost 2 10 H O( n ) in worst case (remember ?) 20 5 8 12 21 � Building an n -node binomial heap using n 9 32 20 31 12 30 successive insertions is O( n ) time complexity 11 13 40 44 � Later we will show that the amortized time of 18 Binomial-Heap-Insert is Θ (1) Minimum : O( log n ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj Binomial Heap : Extract Min. Binomial Heap : Extract Min. 10 10 Union : O( log n ) H H 20 20 5 8 12 21 H * 21 5 9 32 20 31 12 30 H Delete min. root : Θ (1) 8 9 32 20 11 13 40 44 21 12 8 5 H * 10 31 12 11 13 40 18 � the root of B k has degree k 30 31 12 9 32 20 � k is Θ ( log n ) worst case 12 20 44 18 44 11 13 40 � Θ ( log n ) children 30 18 � Reverse list : Θ ( log n ) http://www.cp.eng.chula.ac.th/faculty/spj http://www.cp.eng.chula.ac.th/faculty/spj