An Overview of Transience Bounds in Max-Plus Algebra

We survey and discuss upper bounds on the length of the transient phase of max-plus linear systems and sequences of max-plus matrix powers. In particular, we explain how to extend a result by Nachtigall to yield a new approach for proving such bounds and we state an asymptotic tightness result by using an example given by Hartmann and Arguelles.Comment: 13 pages, 2 figure

Becoming and Building Community in Toronto: 1980 to Present

I left Grenada in December of 1980 to visit Canada. Like many other Caribbean immigrants, when I first came up to Canada I was shocked by the cold and unsure of what my life would look like. Forty years later, I look back at the life I’ve built in Canada and see how my community experiences led me to be a community leader. This is my story, but it is the story of many others as well. I reflect here on the role of women as leaders, the difficult conditions for immigrants, the history of domestic work, and on housing as a human right. I hope my story will explain what policies and actions I have seen that actually help support families, and how we can learn from this for the future

General Transience Bounds in Tropical Linear Algebra via Nachtigall Decomposition

International audienceWe present general transience bounds in tropical linear algebra based on Nachtigall's matrix decomposition. Our approach is also applicable to reducible matrices. The core technical novelty are general bounds on the transient of the maximum of two eventually periodic sequences. Our proof is algebraic in nature, in contrast to the existing purely graph-theoretic approaches

Fast, Robust, Quantizable Approximate Consensus

We introduce a new class of distributed algorithms for the approximate consensus problem in dynamic rooted networks, which we call amortized averaging algorithms. They are deduced from ordinary averaging algorithms by adding a value-gathering phase before each value update. This results in a drastic drop in decision times, from being exponential in the number n of processes to being polynomial under the assumption that each process knows n. In particular, the amortized midpoint algorithm is the first algorithm that achieves a linear decision time in dynamic rooted networks with an optimal contraction rate of 1/2 at each update step. We then show robustness of the amortized midpoint algorithm under violation of network assumptions: it gracefully degrades if communication graphs from time to time are non rooted, or under a wrong estimate of the number of processes. Finally, we prove that the amortized midpoint algorithm behaves well if processes can store and send only quantized values, rendering it well-suited for the design of dynamic networked systems. As a corollary we obtain that the 2-set consensus problem is solvable in linear time in any dynamic rooted network model

Diffusive clock synchronization in highly dynamic networks

International audienceThis paper studies the clock synchronization problem in highly dynamic networks. We show that diffusive synchronization algorithms are well adapted to environments in which the network topology may change unpredictably. In a diffusive algorithm, each node repeatedly (i) estimates the clock difference to its neighbors via broadcast of zero-bit messages, and (ii) updates its local clock according to a weighted average of the estimated differences. The system model allows for drifting local clocks, running at possibly different frequencies. We show that having a rooted spanning tree in the network at every time instance suffices to solve clock synchronization. We do not require any stability of the spanning tree, nor do we impose that the links of the spanning tree be known to the nodes. Explicit bounds on the convergence speed are obtained. In particular, our results settle an open question posed by Simeone and Spagnolini to reach clock synchronization in dynamic networks in the presence of nonzero clock drift. We also identify certain reasonable assumptions that allow for a significant higher convergence speed, e.g., bidirectional networks or random graph models