Binary Search Trees vs. Binary Heaps
Binary Search Tree Condition Given a node i –i.value is the stored object –i.left and i.right point to other nodes All of i’s left children and grand-children are less than i.value All of i’s right children and grand-children are greater than i.value
Binary Search Trees Binary search trees can be easily implemented using arrays X
Binary Search Trees Root is at index 1 (i = 1) Left(i) { return i*2; }Right(i) { return i*2 + 1; } X
Binary Search Trees bool Find_rec(int x, int i) { if (a[i] == -1 ) return false; else if (x == a[i]) return true; else if (x < a[i]) Find_rec(x, i*2); else Find_rec(x, i*2+1); } bool Find(int x) { return Find_rec(x, 1); } …
Binary Search Tree Features Find O(log N) on average Insert in proper order O(log N) on average O(log N) to find the correct location + O(1) to perform insertion Return Min O(log N) on average –Keep moving left until you see a -1 Return Max O(log N) on average –Keep moving left until you see a -1 Print in-order O(N) all the time –See in-order traversal in book
Priority Queue (Min) Three functions –push(x) – pushes the value x into the queue –min() – return the minimum value in the queue –delete() – removes the minimum value in the queue Tons of applications: –OS process queues, Transaction Processing, Packet routing in advanced networks, used in various other algoriothms
Priority Queue (Min) Consider using an un-order Array –push(x) – O(1) just add it to the end of the array. –min() – O(N) sequential search for min value –delete() – O(N) might have to shift entire array
Priority Queue (Min) Consider using an Ordered Array –push(x) – O(N) to find correct location and shift array appropriately –min() – O(1) return the first value –delete() – O(N) might have to shift entire array Recall using an un-order Array –push(x) – O(1) just add it to the end of the array. –min() – O(N) sequential search for min value –delete() – O(N) might have to shift entire array
Priority Queue (Min) Why are simple array implementation bad? O(N) is not a problem, right? Consider this application: –A private network router has 10 million packets coming in every minute (mostly junk, spam, etc.) and I only want to let through the top 1 million (#1 is top priority – min)
Priority Queue (Min) N = 10 million in-coming packets M = 1 million out-going packets (top priority – priority #1) Consider using an Ordered Array –push(x) – O(N) Must do this N times N*N –min() – O(1) –delete() – O(1) Must do this M times M Recall using an un-order Array –push(x) – O(1) Must do this N time (not a problem) –min() – O(N) Must do this M times N*M –delete() – O(N) Must do this M times N*M
N*M N all the packets 10 million M – 1 million top priority packets Must be processed in one minute. Assume your computer can do 10 billion operation per second 600 billion operation in one minute. Unfortunately, N*M is 10 trillion operations.
Priority Queue (Min) Consider using an BST –push(x) – O(log N) add to the correct position –Log(n) * n, n = 10,000,000 –min() – O(log N) return the left-most node –delete() – O(log N) Recall using an un-order Array –push(x) – O(1) just add it to the end of the array. –min() – O(N) sequential search for min value –delete() – O(N) might have to shift entire array
Priority Queue (Min) Is this possible? –push(x) – O(1) to find correct location an shift array appropriately –min() – O(1) return the first value –delete() – O(log N)
Binary Heaps (Min Heap) Given a node i –i.value is the stored object –i.left and i.right point to other nodes All of i’s left children and grand-children are greater than i.value All of i’s right children and grand-children are greater than i.value
Binary Heaps (Min Heap) Binary heaps can be easily implemented using arrays
Binary Heap Features Find O(N) requires sequential search Insert in proper order O(1) on average Amazing Heap Property Return Min O(log N) on average –O(1) to return min –O(log N) to restore the heap property Print in-order O(N log N) requires progressively deleting the min.
Priority Queue (Min) N = 10 million in-coming packets M = 1 million out-going packets (top priority – priority #1) Consider using an Heap –push(x) – O(1) Must do this N times (not a problem) –min() – O(1) –delete() – O(log N) Must do this M times log(N)*M Recall using an un-order Array –push(x) – O(1) Must do this N time (not a problem) –min() – O(N) Must do this M times N*M –delete() – O(N) Must do this M times N*M
log(N)*M N all the packets 10 million M – 1 million top priority packets Must be processed in one minute. Assume your computer can do 1 billion operation per second 60 billion operation in one minute. What is log(N)*M?
Summary: Priority Queue Implementations BST Implementation Push: O(log N) Find Min: O(log N) Remove Min: O(log N) Pushing N = 10,000, million operations Removing M minimums M = 1,000, million operations Binary Heap Implement. Push: O(1) Find Min: O(1) Remove Min: O(log N) Pushing N = 10,000, million operations Removing M minimums M = 1,000, million operations