Introduction Variations of Classical Algorithms Implementation Enhancements The Fastest Algorithms Experiments on Union-Find Algorithms for the Disjoint-Set Data Structure Md. Mostofa Ali Patwary* Jean Blair** Fredrik Manne* *Department of Informatics, University of Bergen, Norway **Department of EE and CS,United States Military Academy, USA May 20-22, 2010 SEA 2010, Italy. Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Implementation Enhancements The Fastest Algorithms Overview ◮ Extensive experimental study comparing 55 different variations of Union-Find algorithm. ◮ The study includes: ◮ All the classical algorithms. ◮ Several recently suggested enhancements. ◮ Different combinations and optimizations of these. ◮ Main Result: A somewhat forgotten simple algorithm developed by M artin R em in 1976 is the fastest algorithm. Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Implementation Enhancements The Fastest Algorithms Related Experimental Studies Reference Application Computing # of Algorithms [Liu, 1990] Sparse matrix Factorization 2 [Gilbert et al., 1994] Sparse matrix Factorization 6 [Wassenberg et al., 2008] Image processing Labeling 8 [Wu et al., 2009] Image processing Labeling 3 [Hynes, 1998] Graphs Connected components 18 [Osipov et al., 2009] Graphs Minimum spanning tree 2 Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Implementation Enhancements The Fastest Algorithms Outline Introduction Applications and Definitions Main Operations (Union-Find) Variations of Classical Algorithms Union Techniques Compression Techniques Interleaved Algorithms Implementation Enhancements 1. Immediate Parent Check [Osipov et al., 2009] 2. Better Interleaved Algorithms [Manne and Patwary, 2009] 3. Memory Efficient Algorithms The Fastest Algorithms Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Disjoint-Set Data Structure: Definitions ◮ U ⇒ set of n elements and S i ⇒ a subset of U . ◮ S 1 and S 2 are disjoint if S 1 ∩ S 2 = ∅ . ◮ Maintains a dynamic collection S 1 , S 2 , . . . S k of disjoint sets which together cover U . ◮ Each set is identified by a representative x . ◮ A set of algorithms that operate on this data structure is often referred to as a Union-Find algorithm. Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Main Operations ◮ Each set is represented by a rooted tree, pointer towards root. a ◮ The element in the root node is d the representative of the set. b ◮ Parent pointer p ( x ) denotes the e c parent of node x . ◮ Two main operations. Figure: S i = { a , b , c , d , e } . ◮ Find ( x ). ◮ Union ( x , y ). Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Find ( x ) a b ◮ To which set does a given element c x belong ⇒ Find ( x ). ◮ Returns the root (representative) d of the set that contain x . e Figure: Find( d ). Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Union ( x , y ) a ◮ Create a new set from the union of f two existing sets containing x and d b g y ⇒ Union ( x , y ). e ◮ Change the parent pointer of one c root to the other one. Figure: Union ( c , g ). Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Union ( x , y ) a ◮ Create a new set from the union of f two existing sets containing x and d b g y ⇒ Union ( x , y ). e ◮ Change the parent pointer of one c root to the other one. Figure: Union ( c , g ). Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Variations of Classical Algorithms Applications and Definitions Implementation Enhancements Main Operations (Union-Find) The Fastest Algorithms Use of Union-Find for Computing Connected Components: G = ( V , E ) 1: S ← ∅ 2: for each x ∈ V do ◮ Note that if the edges are Makeset ( x ) 3: processed by increasing 4: for each ( x , y ) ∈ E do weight then this algorithm is if Find ( x ) � = Find ( y ) then 5: Kruskal’s algorithm. Union ( x , y ) 6: S ← S ∪ { ( x , y ) } 7: Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Union Techniques ◮ Naive-Link ( nl ) ◮ Link-by-Size ( ls ) ◮ Link-by-Rank ( lr ) Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Union Techniques : Link-by-Rank ( lr ) 2 1 ◮ Each set maintains a a f rank value, intially 0. d ◮ Lowest ranked root b g ⇒ higher ranked root. e ◮ Equal ranked roots ⇒ c root of the combined tree is increased by 1. Figure: Union . Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Union Techniques : Link-by-Rank ( lr ) 2 1 ◮ Each set maintains a a f rank value, intially 0. d ◮ Lowest ranked root b g ⇒ higher ranked root. e ◮ Equal ranked roots ⇒ c root of the combined tree is increased by 1. Figure: Union . Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Union Techniques : Link-by-Rank ( lr ) 2 2 ◮ Each set maintains a a f rank value, intially 0. d ◮ Lowest ranked root b g ⇒ higher ranked root. e i h ◮ Equal ranked roots ⇒ c root of the combined tree is increased by 1. Figure: Union . Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Union Techniques : Link-by-Rank ( lr ) 3 2 2 ◮ Each set maintains a a f rank value, intially 0. d ◮ Lowest ranked root b g ⇒ higher ranked root. e i h ◮ Equal ranked roots ⇒ c root of the combined tree is increased by 1. Figure: Union . Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Compression Techniques ◮ Naive-Find ( nf ) ◮ Path-Compression ( pc ) ◮ Path-Splitting ( ps ) ◮ Path-Halving ( ph ) ◮ Type-0-Reversal ( r0 ) ◮ Type-1-Reversal ( r1 ) ◮ Collapsing ( co ) Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Compression Techniques a ◮ Reduce the height of the tree b during the Find operation. ◮ Subsequent Find operations c require less time. d ◮ Find-path of a node x is the path of parent pointers from x upto the e root of the tree. Figure: Find-path( d ). Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
Introduction Union Techniques Variations of Classical Algorithms Compression Techniques Implementation Enhancements Interleaved Algorithms The Fastest Algorithms Compression Techniques : Path-Compression ( pc ) a ◮ Set the parent b pointers of all nodes c in the find path to the root. d ◮ Need to traverse the find-path twice. e Figure: Find ( e ) with pc Md. Mostofa Ali Patwary, Jean Blair, and Fredrik Manne Experiments on Union-Find Algorithms
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