Intro-to-algo (counting sort, radix sort, decis...

======================================== Lecture 5 How fast can we sort? comparison sorts e.g. quicksort, insertion sort, merge sort, heapsort No comparison sorting algorithm could run better than nlgn. ==================================== Decision-tree modal (binary tree) A decision tree can model the execution of any comparison sort: => One tree for each input size n => View the algorith as splitting whenever it compares two elements => tree lists comparsions along all possible instruction traces => running time (# comparsisons) = length of path    worst case running time = the height of the tree (the longest path) Lower bounder on decision tree sorting: Any decision tree sorting n elements must have height omaga(nlgn) ==================================== Sorting in linear time ==================================== Counting sort input: A[1:n], each A[i] belongs to {1, 2, 3, ..., k} output: B[1:n] = sorting of A Auxiliary storage c[1 ... k] for i <- 1 to k     do c[i] <- 0 for j <- 1 to n     do c[a[j]] <- c[a[j]] + 1 //c[i] is the total number of elements whose value equal to i for j <- 2 to n     do c[j] = c[j-1] + c[j] //c[j] is the total number of elements whose value <= j for j <- n downto 1     do B[c[a[j]]] <- A[j] //where A[j] should go in B        c[a[j]] <- c[a[j]] - 1 running time = zita(k) + zita(n) + zita(k) + zita(n) = zita(n+k) if k = zita(n) ==> linear time sorting algorithm if k is relatively small -> good sorting algorithm =================================== stable sort perserve the relative order of equal elements counting sort is a stable sort =================================== Radix sort digit by digit sort least sig -> most sig (must be stable sort) use counting sort as a subroutine ---- use counting sort for each digit (k+n) ---- say n integer, each b bits ---- split each integer b/r digits, each r bits long