A Functional Approach to Standard Binary Heaps

This paper describes a new and purely functional implementation technique of binary heaps. A binary heap is a tree-based data structure that implements priority queue operations (insert, remove, minimum/maximum) and guarantees at worst logarithmic running time for them. Approaches and ideas described in this paper present a simple and asymptotically optimal implementation of immutable binary heap

Functional programming and graph algorithms

This thesis is an investigation of graph algorithms in the non-strict purely functional language Haskell. Emphasis is placed on the importance of achieving an asymptotic complexity as good as with conventional languages. This is achieved by using the monadic model for including actions on the state. Work on the monadic model was carried out at Glasgow University by Wadler, Peyton Jones, and Launchbury in the early nineties and has opened up many diverse application areas. One area is the ability to express data structures that require sharing. Although graphs are not presented in this style, data structures that graph algorithms use are expressed in this style. Several examples of stateful algorithms are given including union/find for disjoint sets, and the linear time sort binsort. The graph algorithms presented are not new, but are traditional algorithms recast in a functional setting. Examples include strongly connected components, biconnected components, Kruskal's minimum cost spanning tree, and Dijkstra's shortest paths. The presentation is lucid giving more insight than usual. The functional setting allows for complete calculational style correctness proofs - which is demonstrated with many examples. The benefits of using a functional language for expressing graph algorithms are quantified by looking at the issues of execution times, asymptotic complexity, correctness, and clarity, in comparison with traditional approaches. The intention is to be as objective as possible, pointing out both the weaknesses and the strengths of using a functional language

Finding the Median in Linear Worst-Case Time

Department of Computer Scienc

Strengthened Lazy Heaps: Surpassing the Lower Bounds for Binary Heaps

Let $n$ denote the number of elements currently in a data structure. An in-place heap is stored in the first $n$ locations of an array, uses $O(1)$ extra space, and supports the operations: minimum, insert, and extract-min. We introduce an in-place heap, for which minimum and insert take $O(1)$ worst-case time, and extract-min takes $O(\lg{} n)$ worst-case time and involves at most $\lg{} n + O(1)$ element comparisons. The achieved bounds are optimal to within additive constant terms for the number of element comparisons. In particular, these bounds for both insert and extract-min -and the time bound for insert- surpass the corresponding lower bounds known for binary heaps, though our data structure is similar. In a binary heap, when viewed as a nearly complete binary tree, every node other than the root obeys the heap property, i.e. the element at a node is not smaller than that at its parent. To surpass the lower bound for extract-min, we reinforce a stronger property at the bottom levels of the heap that the element at any right child is not smaller than that at its left sibling. To surpass the lower bound for insert, we buffer insertions and allow $O(\lg^2{} n)$ nodes to violate heap order in relation to their parents

Compiling Irregular Software to Specialized Hardware

High-level synthesis (HLS) has simplified the design process for energy-efficient hardware accelerators: a designer specifies an accelerator’s behavior in a “high-level” language, and a toolchain synthesizes register-transfer level (RTL) code from this specification. Many HLS systems produce efficient hardware designs for regular algorithms (i.e., those with limited conditionals or regular memory access patterns), but most struggle with irregular algorithms that rely on dynamic, data-dependent memory access patterns (e.g., traversing pointer-based structures like lists, trees, or graphs). HLS tools typically provide imperative, side-effectful languages to the designer, which makes it difficult to correctly specify and optimize complex, memory-bound applications. In this dissertation, I present an alternative HLS methodology that leverages properties of functional languages to synthesize hardware for irregular algorithms. The main contribution is an optimizing compiler that translates pure functional programs into modular, parallel dataflow networks in hardware. I give an overview of this compiler, explain how its source and target together enable parallelism in the face of irregularity, and present two specific optimizations that further exploit this parallelism. Taken together, this dissertation verifies my thesis that pure functional programs exhibiting irregular memory access patterns can be compiled into specialized hardware and optimized for parallelism. This work extends the scope of modern HLS toolchains. By relying on properties of pure functional languages, our compiler can synthesize hardware from programs containing constructs that commercial HLS tools prohibit, e.g., recursive functions and dynamic memory allocation. Hardware designers may thus use our compiler in conjunction with existing HLS systems to accelerate a wider class of algorithms than before

QuickHeapsort: Modifications and improved analysis

We present a new analysis for QuickHeapsort splitting it into the analysis of the partition-phases and the analysis of the heap-phases. This enables us to consider samples of non-constant size for the pivot selection and leads to better theoretical bounds for the algorithm. Furthermore we introduce some modifications of QuickHeapsort, both in-place and using n extra bits. We show that on every input the expected number of comparisons is n lg n - 0.03n + o(n) (in-place) respectively n lg n -0.997 n+ o (n). Both estimates improve the previously known best results. (It is conjectured in Wegener93 that the in-place algorithm Bottom-Up-Heapsort uses at most n lg n + 0.4 n on average and for Weak-Heapsort which uses n extra-bits the average number of comparisons is at most n lg n -0.42n in EdelkampS02.) Moreover, our non-in-place variant can even compete with index based Heapsort variants (e.g. Rank-Heapsort in WangW07) and Relaxed-Weak-Heapsort (n lg n -0.9 n+ o (n) comparisons in the worst case) for which no O(n)-bound on the number of extra bits is known