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The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.

We study the problem of sorting N elements in presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability p, but repeating the same comparison gives always the same result. The best known algorithms for this problem have running time O(N^2) and achieve an optimal maximum dislocation of O(log N) for constant error probability. Note that no algorithm can achieve dislocation o(log N), regardless of its running time. In this work we present the first subquadratic time algorithm with optimal maximum dislocation: Our algorithm runs in tilde{O}(N^{3/2}) time and guarantees O(log N) maximum dislocation with high probability. Though the first version of our algorithm is randomized, it can be derandomized by extracting the necessary random bits from the results of the comparisons (errors).

We study the problem of sorting N elements in presence of persistent errors in comparisons: In this classical model, each comparison between two elements is wrong independently with some probability p, but repeating the same comparison gives always the same result. The best known algorithms for this problem have running time O(N^2) and achieve an optimal maximum dislocation of O(log N) for constant error probability. Note that no algorithm can achieve dislocation o(log N), regardless of its running time. In this work we present the first subquadratic time algorithm with optimal maximum dislocation: Our algorithm runs in tilde{O}(N^{3/2}) time and guarantees O(log N) maximum dislocation with high probability. Though the first version of our algorithm is randomized, it can be derandomized by extracting the necessary random bits from the results of the comparisons (errors).

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.

We present a sorting algorithm for the case of recurrent random comparison errors. The algorithm essentially achieves simultaneously good properties of previous algorithms for sorting n distinct elements in this model. In particular, it runs in O(n^2) time, the maximum dislocation of the elements in the output is O(log n), while the total dislocation is O(n). These guarantees are the best possible since we prove that even randomized algorithms cannot achieve o(log n) maximum dislocation with high probability, or o(n) total dislocation in expectation, regardless of their running time.

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before.

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before.

The conventional wisdom of fiscal decentralization advocates is that fiscal decentralization can facilitate the economic development of a country. In addition, the World Bank and IMF have identified over sixty countries where decentralization is an important element of development strategy. However, with the proliferating implementation of fiscal decentralization, the actual outcome has varied from country to country. A major barrier to understanding is the lack of well-defined theoretical framework to empirically measure fiscal decentralization in a policy relevant way. The most widely used measurement is the ratio of sub-national government expenditure/revenue to total government expenditure/revenue. But this indicator is criticized too simple to capture the dimensions of fiscal decentralization. Specifically, it ignores key qualitative dimensions, such as taxing power, borrowing power and the independence of local officials, which are also very important to implementing fiscal decentralization. The primary contribution of this dissertation is a novel theoretical framework for empirical measurement of fiscal decentralization. The research question is: what types of fiscal decentralization system produce better economic performance? We go beyond the traditional way of measuring a country's fiscal decentralization, treating it instead as a system with its own institutional design. Adopting this method requires us investigate the institutional arrangement for fiscal decentralization in a country. The institutional arrangement we review in detail includes: supra-national government, federal or unitary state, numbers of tiers of governments, taxing power, borrowing power, and independent local official. These components have also been recognized by many economists and policy analysts. What is original to this work is that, after identifying these institutional arrangements, we can group different countries with similar institutionally similar fiscal decentralization sy