A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

An in-place algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others...

. A variant of Heapsort---named Ultimate Heapsort---is presented that sorts n elements in-place in \Theta(n log 2 (n+ 1)) worst-case time by performing at most n log 2 n + \Theta(n) key comparisons and n log 2 n + \Theta(n) element moves. The secret behind Ultimate Heapsort is that it occasionally transforms the heap it operates with to a two-layer heap which keeps small elements at the leaves. Basically, Ultimate Heapsort is like Bottom-Up Heapsort but, due to the two-layer heap property, an element taken from a leaf has to be moved towards the root only O(1) levels, on an average. Let a[1::n] be an array of n elements each consisting of a key and some information associated with this key. This array is a (maximum) heap if, for all i 2 f2; : : : ; ng, the key of element a[bi=2c] is larger than or equal to that of element a[i]. That is, a heap is a pointer-free representation of a left complete binary tree, where the elements stored are partially ordered according to their keys. Ele...

Given a set S with real-valued members, associated with each member one of two possible types; a multi-partitioning of S is a sequence of the members of S such that if x, y ∈ S have different types and x < y, x precedes y in the multi-partitioning of S. We give two distribution-sensitive algorithms for the set multi-partitioning problem and a matching lower bound in the algebraic decision-tree model. One of the two algorithms can be made stable and can be implemented in place. We also give an output-sensitive algorithm for the problem.

We establish the first nontrivial lower bounds on timespace tradeoffs for the selection problem. We prove that any comparison-based randomized algorithm for finding the median requires Ω(n log logS n) expected time in the RAM model (or more generally in the comparison branching program model), if we have S bits of extra space besides the read-only input array. This bound is tight for all S ≫ log n, and remains true even if the array is given in a random order. Our result thus answers a 16-year-old question of Munro and Raman, and also complements recent lower bounds that are restricted to sequential access, as in the multi-pass streaming model [Chakrabarti et al., SODA 2008]. We also prove that any comparison-based, deterministic, multi-pass streaming algorithm for finding the median requires Ω(n log ∗ (n/s) + n log s n) worst-case time (in scanning plus comparisons), if we have s cells of space. This bound is also tight for all s ≫ log 2 n. We get deterministic lower bounds for I/O-efficient algorithms as well. All proofs in this paper involve “elementary ” techniques only. 1

It is shown that an array of n elements can be sorted using O(1) extra space, O(n log n= log log n) element moves, and n log 2 n+O(n log log n) comparisons. This is the first in-place sorting algorithm requiring o(n log n) moves in the worst case while guaranteeing O(n log n) comparisons but, due to the constant factors involved, the algorithm is predominantly of theoretical interest. Key words: In-place algorithms, sorting, merging, mergesort, multiway merge 1