Colocation, Colocation, Colocation: Optimizing Placement in the Hybrid Cloud

• Published in: Algorithmic Aspects of Cloud Computing pp. 25 - 45
• Main Authors: Aiyar, Srinivas, Gupta, Karan, Rajaraman, Rajmohan, Shen, Bochao, Sun, Zhifeng, Sundaram, Ravi
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing 28.04.2019
• Series: Lecture Notes in Computer Science
• ISBN: 9783030197582; 3030197581
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-19759-9_3

Today’s enterprise customer has to decide how to distribute her services among multiple clouds - between on-premise private clouds and public clouds - so as to optimize different objectives, e.g., minimizing bottleneck resource usage, maintenance downtime, bandwidth usage or privacy leakage. These use cases motivate a general formulation, the uncapacitated (A defining feature of clouds is their elasticity or ability to scale with load) multidimensional load assignment problem - VITA(F) (Vectors-In-Total Assignment): the input consists of n, d-dimensional load vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{V} = \{\bar{V}_i | 1\le i \le n\}$$\end{document}, m cloud buckets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B = \{B_j | 1\le j \le m\}$$\end{document} with associated weights \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_j$$\end{document} and assignment constraints represented by a bipartite graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(\bar{V} \cup B, E \subseteq \bar{V} \times B)$$\end{document} restricting load \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{V}_i$$\end{document} to be assigned only to buckets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_j$$\end{document} with which it shares an edge (In a slight abuse of notation, we let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_j$$\end{document} also denote the subset of vectors assigned to bucket \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_j$$\end{document}). F can be any operator mapping a vector to a scalar, e.g., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max $$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\min $$\end{document}, etc. The objective is to partition the vectors among the buckets, respecting assignment constraints, so as to achieve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \min [ \sum _j w_j*F (\sum _{\bar{V}_i \in B_j} \bar{V}_i)] \end{aligned}$$\end{document}We characterize the complexity of VITA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\min )$$\end{document}, VITA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\max )$$\end{document}, VITA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\max - \min )$$\end{document} and VITA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2^{nd}\max )$$\end{document} by providing hardness results and approximation algorithms, LP-Approx involving clever rounding of carefully crafted linear programs. Employing real-world traces from Nutanix, a leading hybrid cloud provider, we perform a comprehensive comparative evaluation versus three natural heuristics - Conservative, Greedy and Local-Search. Our main finding is that on real-world workloads too, LP-Approx outperforms the heuristics, in terms of quality, in all but one case.