elements. The algorithm can be implemented by depth-first search of a binary tree: each level in the tree corresponds to an input number; the left branch corresponds to excluding the number from the set, and the right branch corresponds to including the number (hence the name Inclusion-Exclusion). The memory required is . The run-time can be improved by several heuristics: Process the input numbers in descending order. If the integers included in a given node exceed the sum of the best subset found so far, the node is pruned. If the integers included in a given node, plus all remaining integers, are less than the sum of the best subset found so far, the node is pruned. Horowitz and Sahni In 1974, Horowitz and Sahni published a faster exponential-time algorithm, which runs in time , but requires much more space - . The algorithm splits arbitrarily the n elements into two sets of each. For each of these two sets, it stores a list of the sums of all possible subsets of its elements. Each of these two lists is then sorted. Using even the fastest comparison sorting algorithm, Mergesort for this step would take time . However, given a sorted list of sums for elements, the list can be expanded to two sorted lists with the introduction of a ()th element, and these two sorted lists can be merged in time . Thus, each list can be generated in sorted form in time . Given the two sorted lists, the algorithm can check if an element of the first array and an element of the second array sum up to T in time . To do that, the algorithm passes through the first array in decreasing order (starting at the largest element) and the second array in increasing order (starting at the smallest element). Whenever the sum of the current element in the first array and the current element in the second array is more than T, the algorithm moves to the next element in the first array. If it is less than T, the algorithm moves to the next element in the second array. If two elements that sum to T are found, it stops. (The sub-problem for two elements sum is known as two-sum.) Schroeppel and Shamir In 1981, Schroeppel and Shamir presented an algorithm based on Horowitz and Sanhi, that requires similar runtime - , much