Trees, Glorious Trees. From one new year to another new year, and from one new academic year to another new academic year ראש השנה לאילנות

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1 Trees, Glorious Trees From one new year to another new year, and from one new academic year to another new academic year ראש השנה לאילנות There all kinds of trees: Apple trees Palm trees Bonsai trees and so on... There are also many kinds of binary trees and such: Species Algorithms Floyd-Warshal Dijkstra Kruskal DFS/BFS/BestFS/etc Union-Find (path compression) Huffman A* rooted/unrooted trees binary/ternary/quad trees splay trees parse trees electrical nets decision trees game trees Applications Shortest path in graphs Minimal cost connecting Facility placement

2 Some Notation קיים There exist לכל For all כך ש... that Such קבוצה חלקית Subset ערך "רצפה" x Floor ערך "גג" x Ceiling 3.8 =3 3.8 =4 שייך ל... of Member

3 Binary Search Tree D C F A E H B G I Label: B I G C H E F A D Left: Right: Root: [9]

4 Simple rotation B A A C a1 B C a1 a2 a2

5 Heap Complete Binary tree, but usual implementation uses an Array A[] where: Left(A[i]) = A[2i] Right(A[i]) = A[2i+1] Root = A[1] Parent(A[i]) = A[ i/2 ] Heap property: A[i] Parent(A[i]) Important Algorithms: Heapify(A,i) Build_Heap(A) HeapSort Heap_Insert(A, value) Heapify(A,i) i x y z Heap Heap i 2i 2i+1 A: x y z Heap property OK!

6 Maintaining the heap property

7 Variation of Heapify Lower # of comparisons Keep Larger Child Pointers (LCP) (bit vector) Unique LCP path from root to one leaf "Binary Insert" on this path instead of "float down" Update LCP after exchanges Complexity: Regular 2log(n) operations Variation log(n)+loglog(n)

8 Priority Queues Head Insert, but Move priority forward Operations: Insert Max Extract-Max Applications Scheduling jobs by priority BEZEQ protectzia Minimum Spanning Tree algorithm (MST)

9 Minimum Spanning Tree MST G= (V,E) graph w(u,v) weight of (u,v) E Find tree T E connecting all vertices (Spanning V) such that, Min w(t) = w(u,v) (u,v) T Notes: 1. G is connected G has a spanning tree 2. T = V T is acyclic Motivation for MST: Connect all "pins", "cities", "nodes" with cheapest/shortest total amount of "wire", "roads", "cost". "General" MST algorithm a greedy strategy: 1. T 2. while T not spanning ( T < V -1) 3. do choose safe edge (v,u) to add (T remains acyclic) 4. T T {(v,u)} end do 5. return T

10 Prim s Algorithm for MST (Army ground force attack) - T is an expanding Tree (starting from a chosen vertex as root ) - uses priority queue for the vertices not in the tree - Key(v) is cost of the cheapest "safe" edge which will extend the tree by adding a vertex v. The safe edge is (v, (v)) - (v) is the (eventual) parent of v in T. -

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12 Kruskal s Algorithm for MST (Air force attack) - F is an expanding forest - Can use priority queue (reverse heap) - Complexity O( E log E ) KRUSKAL(G): 1 A = 2 for each v V(G) do 3 MAKE-SET(v) end-do 4 for each (u, v) ordered by increasing weight w(u, v) do 5 if FIND-SET(u) FIND-SET(v): 6 A = A {(u, v)} 7 UNION(u, v) end-do 8 return A

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14 Data structures for disjoint sets S = { S1, S2,..., Sk } Si Sj = i j Representative - unique ri Si Find-Set(x) returns pointer to the unique representative of the set containing x. Union(x,y) combines their two sets, i.e., Sx Sy and assigns its representative. M A R T I N C H U K Y K A M C U H N T R I Y Find-Set Union O(1) O(length of shorter list) Complexity: O(m+n*logn) S =n ; m= #of union, finds Union by concatenating smaller list to end of longer list

15 Forest of trees M C Representative is the root A R H U T I N K Y Find-Set Union O(distance from root) O(1) Union by making root of shorter tree a child of root of longer tree

16 Path compression d d c x b c a b a x Complexity (with path compression) m = #Union + finds n = S O(m * (m,n) ) (m,n) 4, if n estimated number of atoms in the universe

17 Path compression P[x] parent rank[x] > height of x (leaf to x) equality for all roots Make-Set(x) P[x] x rank[x] 0 Union(x, y) r Find_Set(x) s Find_Set(x) if (r s) Link(r,s) Link(r, s) if (rank (r) > rank (s)) then P[s] r else P[r] s if (rank(r) == rank(s) then rank(s) rank(s)+1 Find_Set(x) if(x P[x]) then P[x] Find_Set(P[x]) return P[x]

18 SUMMARY (Prim vs Kruskal) Prim T is an expanding Tree until all vertices are in T Complexity: O(m log n) or O(m + n log n), depending on the data-structures used. Kruskal F is an expanding forest until all subtrees of F form one tree Complexity: O(m log n) depending on the data-structures used. -

19 Single-Source Shortest Paths: Dijkstra Algorithm For a weighted graph G = (V, E, w), Find the shortest paths from a chosen vertex v to all other vertices in V. Google Maps, Waze, MapQuest, Mobi, Yahoo Maps, MyRouteOnline, Route4Me, Dijkstra's algorithm is somewhat similar to Prim's algorithm, or a general search algorithm, in that It maintains a set of nodes for which the shortest paths are already known, then It grows this set based on the node closest to source using one of the nodes in the current shortest path set.

20 EXAMPLE..\barcelona summer school\04demo-dijkstra-1.ppt

21 Other algorithms and techniques you should know: MAXFLOW maximum flow in a weighted network with constraints BFS Breadth First Search DFS Depth First Search NPC NP-completeness and problem reductions How do I search thee, oh graph? Let me count the ways... LexBFS, MCS, A*, IDA*, SMA*, LexDFS,..

22 Planar Graphs Those which can be drawn on the Euclidean plane with edges crossing only at the vertices. Example:

23 Graph Coloring (of vertices) A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible.

24 Graph Coloring (of edges) A 3-edge-coloring of the Desargues graph.

25 Algorithms for Graph Coloring Structured Graphs Often have polynomial-time algorithms General graphs NP-Hard -- so heuristics are usually needed exploit the structure Who is Daniel Brélaz?

26 Bipartite Graphs G(V,E) is Bipartite: V = X Y, E X Y x, x' X xx' E y, y' Y yy' E X Y Theorem: G is Bipartite G is 2 - colorable G has no odd length cycles Greedy Constraint Propagation Algorithm

27 Representation of graphs Graph G = (V, E) Adjacency set Adj(v) = { w V (v,w) E } (Neighborhood) Example: List: Adj(1) : Adj(2) : 2 3 Adj(3) : 4 2 Adj(4) : 1 Matrix:

28 Complexity What to count How to count How accurately to count - Exact - Average - Worst - Probabilistic - Amortized Basic operations - Estimation - Assumptions

29 Asymptotic notation Big Oh Big Omega BigTheta O Θ T(n) and f(n) are positive integer valued Upper bound T(n) is O(f(n)) if c, N>0 T(n) c f(n) n N Lower bound T(n) is (f(n)) if c, N>0 c f(n) T(n) n N Tight bound T(n) is Θ(f(n)) if c1, c2, N>0 n N, c1*f(n) T(n) c2*f(n)

30 More Notations O(g(n)) = O(f(n)) g(n) is O(f(n)) and f(n) is O(g(n)) i.e. g(n) is (f(n)), f(n) is (g(n)) O(g(n)) < O(f(n)) g(n) is O(f(n)) but f(n) is not O(g(n)) Same as "little" o : g(n) is o(f(n))

31 Fraction Theorem O(g(n)) > O(f(n)) limit _g(n)_ n f(n) = c > 0 O(g(n)) = O(f(n)) 0 O(g(n)) < O(f(n)) L'Hopital's Rule (taking derivatives) If limit f(n) = n then limit g(n) = n 0 limit n _g(n)_ f(n) = limit n _g'(n)_ f'(n)

32 For solving recurrences: The "Master" Theorem T(n) = a T(n/b) + f(n) Divide and conquer a 1 b > 1 #subproblems a size of subproblems n/b (n log b a ) a>b T(n) = (n*logn) a=b (f(n)) a<b, a*f(n/b) c*f(n), c<1 T(n) = (n log b a ) if f(n)=o(n log b a- )) e.g. a>b T(n) = (n log b a * log(n)) if f(n) = (n log b a )) e.g. a=b; f(n) = (n) a=1, b=2; f(n) = (1) T(n) = (f(n)) if f(n) = (n log b a+ )) e.g. a<b and a*f(n/b) c*f(n) c<1 almost everywhere

33 Binary Merge Sort T(n) = 2T(n/2) + O(n) = (n*logn) Using "Master" theorem Strassen's Matrix multiply T(n) = 7T(n/2) + O(n 2 ) = (n log 2 7 ) = (n 2.81 ) binary search T(n) = T(n/2) + O(1) = (logn)

34 Everything else you need to know you learned in Kindergarten: w/ A standard algorithms course by Prof. Adam Smith using the Kleinberg and Tardos book VERY GOOD SLIDES. A standard algorithms course by Prof. John W. Byers using the Kleinberg and Tardos book Lectures 5,6,7 give Prim, Kruskal, Dijkstra NICE SLIDES. lectures/ss07/graph_algo A standard full semester Graph Theory with Algorithms course by Prof. Dr. Angelika Steger, Dr. Martin Marciniszyn Algorithm Design Introduction to Algorithms by Jon Kleinberg by Thomas H. Cormen, Charles E. Leiserson, and Éva Tardos Ronald L. Rivest and Clifford Stein