Implementing Todays lecture: the UNION-FIND ADT Basic - PDF document

10/27/2016 The plan Last lecture: • Disjoint sets CSE373: Data Structures and Algorithms • The UNION-FIND ADT for disjoint sets Implementing Today’s lecture: the UNION-FIND ADT • Basic implementation of the UNION-FIND ADT with “up trees” • Optimizations that make the implementation much faster Steve Tanimoto Autumn 2016 This lecture material represents the work of multiple instructors at the University of Washington. Thank you to all who have contributed! Autumn 2016 CSE 373: Data Structures & Algorithms 2 Union-Find ADT Implementation – our goal • Given an unchanging set S , create an initial partition of a set • Start with an initial partition of n subsets – Typically each item in its own subset: {a}, {b}, {c}, … – Often 1-element sets, e.g., {1}, {2}, {3}, …, { n } – Give each subset a “name” by choosing a representative element • May have m find operations • May have up to n -1 union operations in any order • Operation find takes an element of S and returns the – After n -1 union operations, every find returns same 1 set representative element of the subset it is in • Operation union takes two subsets and (permanently) makes one larger subset – A different partition with one fewer set – Affects result of subsequent find operations – Choice of representative element up to implementation Autumn 2016 CSE 373: Data Structures & Algorithms 3 Autumn 2016 CSE 373: Data Structures & Algorithms 4 Up-tree data structure Find find ( x ): • Tree with: – No limit on branching factor – Assume we have O (1) access to each node • Will use an array where index i holds node i – References from children to parent – Start at x and follow parent pointers to root • Start with forest of 1-node trees – Return the root 1 2 3 4 5 6 7 1 3 7 • Possible forest after several unions: find (6) = 7 – Will use roots for 7 1 3 5 4 2 set names 5 4 2 6 6 Autumn 2016 CSE 373: Data Structures & Algorithms 5 Autumn 2016 CSE 373: Data Structures & Algorithms 6 1

10/27/2016 Union Simple implementation union ( x,y ): • If set elements are contiguous numbers (e.g., 1,2,…, n ), use an – Assume x and y are roots array of length n called up • Else find the roots of their trees – Starting at index 1 on slides – Assume distinct trees (else do nothing) – Put in array index of parent, with 0 (or -1, etc.) for a root – Change root of one to have parent be the root of the other • Example: 1 2 3 4 5 6 7 • Notice no limit on branching factor 1 2 3 4 5 6 7 up 0 0 0 0 0 0 0 • Example: 1 2 3 4 5 6 7 1 3 7 1 3 7 up 0 1 0 7 7 5 0 union (1,7) 5 4 2 2 5 4 6 • If set elements are not contiguous numbers, could have a 6 separate dictionary to map elements (keys) to numbers (values) Autumn 2016 CSE 373: Data Structures & Algorithms 7 Autumn 2016 CSE 373: Data Structures & Algorithms 8 Implement operations Two key optimizations // assumes x in range 1,n // assumes x,y are roots int find(int x) { void union(int x, int y){ 1. Improve union so it stays O(1) but makes find O ( log n ) while(up[x] != 0) { up[y] = x; – So m find s and n -1 union s is in O ( m log n + n ) x = up[x]; } – Union-by-size: connect smaller tree to larger tree } return x; } 2. Improve find so it becomes even faster – Make m find s and n -1 union s almost in O ( m + n ) 1 3 7 – Path-compression: connect directly to root during finds 1 2 3 4 5 6 7 5 4 2 up 0 1 0 7 7 5 0 6 • Worst-case run-time for union ? O (1) • Worst-case run-time for find ?  (n)  (m*n) • Worst-case run-time for m find s and n -1 union s? Autumn 2016 CSE 373: Data Structures & Algorithms 9 Autumn 2016 CSE 373: Data Structures & Algorithms 10 The bad case to avoid Union-by-size … 1 2 3 n Union-by-size: … – Always point the smaller (total # of nodes) tree to the root of union (2,1) the larger tree 2 3 n … 1 union (3,2) union (1,7) 3 n : 1 3 7 . 4 1 2 2 n union ( n , n -1) 1 2 5 4 3 2 6 find (1) = n steps!! 1 Autumn 2016 CSE 373: Data Structures & Algorithms 11 Autumn 2016 CSE 373: Data Structures & Algorithms 12 2

10/27/2016 Union-by-size Array implementation Union-by-size: Keep the size (number of nodes in a second array) – Always point the smaller (total # of nodes) tree to the root of – Or have one array of objects with two fields the larger tree 1 3 7 4 1 2 1 2 3 4 5 6 7 0 1 0 7 7 5 0 union (1,7) 5 4 up 2 weight 2 1 4 1 3 7 6 6 1 2 5 4 3 7 1 6 1 1 2 3 4 5 6 7 7 1 0 7 7 5 0 5 4 up 6 2 weight 1 6 6 Autumn 2016 CSE 373: Data Structures & Algorithms 13 Autumn 2016 CSE 373: Data Structures & Algorithms 14 Nice trick The Bad case? Now a Great case… Actually we do not need a second array… – Instead of storing 0 for a root, store negation of size union (2,1) 1 2 3 n – So up value < 0 means a root … 1 3 7 union (3,2) 4 2 3 n 1 2 1 2 3 4 5 6 7 : up -2 1 -1 7 7 5 -4 1 … 5 4 2 2 n union ( n , n -1) 6 1 3 2 1 3 7 6 1 1 2 3 4 5 6 7 … find (1) constant here 1 3 n up 7 1 -1 7 7 5 -6 5 4 2 6 Autumn 2016 CSE 373: Data Structures & Algorithms 15 Autumn 2016 CSE 373: Data Structures & Algorithms 16 General analysis Exponential number of nodes P( h )= With union-by-size, up-tree of height h has at least 2 h nodes • Showing one worst-case example is now good is not a proof that the worst-case has improved Proof by induction on h … Base case: h = 0: The up-tree has 1 node and 2 0 = 1 • So let’s prove: • – union is still O (1) – this is “obvious” • Inductive case: Assume P( h ) and show P( h +1) – find is now O ( log n ) – A height h +1 tree T has at least one height h child T1 – T1 has at least 2 h nodes by induction • Claim: If we use union-by-size, an up-tree of height h has at – And T has at least as many nodes not in T1 than in T1 least 2 h nodes • Else union-by-size would have T – Proof by induction on h … had T point to T1, not T1 point to T (!!) – So total number of nodes is at least 2 h + 2 h = 2 h+1 h T1 . Autumn 2016 CSE 373: Data Structures & Algorithms 17 Autumn 2016 CSE 373: Data Structures & Algorithms 18 3

10/27/2016 The key idea The new worst case Intuition behind the proof: No one child can have more than half the n/2 Unions-by-size nodes T h T1 n/4 Unions-by-size So, as usual, if number of nodes is exponential in height, then height is logarithmic in number of nodes So find is O ( log n ) Autumn 2016 CSE 373: Data Structures & Algorithms 19 Autumn 2016 CSE 373: Data Structures & Algorithms 20 The new worst case (continued) What about union-by-height We could store the height of each root rather than size After n/2 + n/4 + …+ 1 Unions-by-size: • Still guarantees logarithmic worst-case find – Proof left as an exercise if interested log n • But does not work well with our next optimization – Maintaining height becomes inefficient, but maintaining size still easy Worst Height grows by 1 a total of log n times find Autumn 2016 CSE 373: Data Structures & Algorithms 21 Autumn 2016 CSE 373: Data Structures & Algorithms 22 Two key optimizations Path compression • Simple idea: As part of a find , change each encountered 1. Improve union so it stays O(1) but makes find O ( log n ) node’s parent to point directly to root – So m find s and n -1 union s is O ( m log n + n ) – Faster future find s for everything on the path (and their – Union-by-size: connect smaller tree to larger tree descendants) 1 7 2. Improve find so it becomes even faster 7 1 – Make m find s and n -1 union s almost O ( m + n ) find (3) – Path-compression: connect directly to root during finds 5 2 3 6 5 4 4 2 11 12 10 8 9 6 8 9 3 10 11 12 Autumn 2016 CSE 373: Data Structures & Algorithms 23 Autumn 2016 CSE 373: Data Structures & Algorithms 24 4