analysis is similar to that for quicksortand the time complexity is ( log nbecause of off-by-one errorsit takes some time to get the program just right now we will solve this problem by using dp natural thought is to assume we have the solution for the subarray [ howevereven if we knew the largest sum subarray for subarray [ ]it does not help us solve the problem for [ instead we need to know the subarray amongst all subarrays [ ] with the smallest subarray sum the desired value is [ minus this subarray' sum we compute this value by iterating through the array for each index jthe maximum subarray sum ending at is equal to [jmink< [kduring the iterationwe track the minimum [kwe have seen so far and compute the maximum subarray sum for each index the time spent per index is constantleading to an (ntime and ( space solution it is legal for all array entries to be negativeor the array to be empty the algorithm handles these input cases correctly def find_maximum_subarray ( )min_sum max_sum for running_sum in itertools accumulate ( )min_sum min(min_sum running_sum max_sum max(max_sum running_sum min_sum return max_sum dynamic programming boot camp the programs presented in the introduction for computing the fibonacci numbers and the maximum subarray sum are good illustrations of dp count the number of ways to traverse array in this problem you are to count the number of ways of starting at the top-left corner of array and getting to the bottom-right corner all moves must either go right or down for examplewe show three ways in array in figure (as we will seethere are total of possible ways for this example figure paths through array write program that counts how many ways you can go from the top-left to the bottom-right in array hintif and you can get to (ijfrom ( jor