We consider the indexable dictionary problem, which consists of storing a set S ⊆ {0, . . . , m − 1} for some integer m while supporting the operations of rank(x ), which returns the number of elements in S that are less than x if x ∈ S, and −1 otherwise; and select(i ), which returns the ith smallest element in S. We give a data structure that supports both operations in O(1) time on the RAM model and requires B(n, m) + o(n) + O(lg lg m) bits to store a set of size n, where B(n, m) = lg m n is the minimum number of bits required to store any n-element subset from a universe of size m. Previous dictionaries taking this space only supported (yes/no) membership queries in O(1) time. In the cell probe model we can remove the O(lg lg m) additive term in the space bound, answering a question raised by Fich and Miltersen [1995] and Pagh [2001].We present extensions and applications of our indexable dictionary data structure, including:-an information-theoretically optimal representation of a k-ary cardinal tree that supports standard operations in constant time; -a representation of a multiset of size n from {0, . . . , m − 1} in B(n, m + n) + o(n) bits that supports (appropriate generalizations of) rank and select operations in constant time; and + O(lg lg m)-a representation of a sequence of n nonnegative integers summing up to m in B(n, m + n) + o(n) bits that supports prefix sum queries in constant time. ACM Reference Format:Raman, R., Raman, V., and Rao, S. S. 2007. Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans.

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This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or k-ary tries), where each node has k slots, labelled {1, . . . , k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time. For ordinal trees we support the operations of finding the degree, parent, ith child, and subtree size. For cardinal trees the structure also supports finding the child labelled i of a given node apart from the ordinal tree operations. These representations also provide a mapping from the n nodes of the tree onto the integers {1, . . . , n}, giving unique labels to the nodes of the tree. This labelling can be used to store satellite information with the nodes efficiently.

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We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) r-Domination problem and the (Connected) p-Center problem for unweighted and undirected graphs. Given a graph G, we show how to construct a (connected) r + O(µ)-dominating set D with |D| ≤ |D * | efficiently. Here, D * is a minimum (connected) r-dominating set of G and µ is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of G. Additionally, we show that a +O(µ)-approximation for the (Connected) p-Center problem on G can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks , social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.