# Data Management in Hierarchical Bus Networks

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4 which is a contradiction to the assertion that a placement with congestion at most 4k exists. 3. THE ETENDED-NIBBLE STRATEGY Assume we are given a hierarchical bus network T = (P B, E, b), a set of shared objects, and read and write frequencies h r, h w : P N. Our exted-nibble strategy consists of the following three steps. Step 1: the nibble strategy Compute an optimal placement of the shared data objects under the assumption that also inner nodes may hold copies. This is shown in [10] to be possible by a simple, linear time algorithm, the nibble strategy. Step 2: the deletion algorithm removing rarely used copies For x let κ x := P P P hw(p, x) denote the write contention induced by x. Modify the above placement so that all copies of x serve at least κ x read or write requests. We will show that this is possible such that the congestion is at most doubled. Step 3: the mapping algorithm move copies from inner nodes to leaves Replace each copy stored on an inner node by one or more copies on leaves. We show how to do this such that the placement we get produces a congestion that is at most a factor of 7 away from optimal. Computing this strategy needs sequential time O( P B height(t ) log(degree(t ))), and can be executed in a distributed way on T in time O( P B log(degree(t )) + height(t )). The remainder of this chapter describes the three steps in more detail. In the following chapter the analysis of the algorithm is given. 3.1 Step 1: the nibble strategy In this section we present the nibble strategy that was introduced in [10]. Note that the nibble strategy places copies also on inner nodes. Therefore, we do not distinguish between processors and buses in the following, i.e., we are given an ordinary tree T = (V, E) with V = P B and E = E. The nibble strategy places each data object x indepently from the other objects. We use the following notations and definitions concerning the access rates to a fixed object x. For a node v (processor or bus), define r(v) = h r(v, x), w(v) = h w(v, x), and h(v) = r(v) + w(v). Thus, h(v) represents the total number of accesses to object x by node v. h(v) is also denoted the weight of node v. For any subtree T = (V, E ), i.e., a connected subgraph of T, we define r(t ) = P v V r(v), w(t ) = P v V w(v), and h(t ) = r(t ) + w(t ). All copies of x are placed on some nodes that form a connected component including the center of gravity g(t ), which is defined as follows. Consider the set of nodes v such that the removal of v partitions T into subtrees, each of which contains at most 1/2 of the total weight. It is easy to check that this set is not empty. We choose an arbitrary node from this set to be the center of gravity g(t ), e.g., the one with the smallest index. Note that the gravity center deps on the access rates to object x such that the gravity center for another object possibly is a different node. In the following, g(t ) is assumed to be the root of the tree T, which defines the parent and the children of each node. The subtree T (v) rooted at v V is defined to be the maximal subtree including v but not the parent of v. After we have fixed this notation, the rest of the description of the nibble strategy is very simple. The nibble placement: A node v gets a copy of x if and only if v = g(t ) or h(t (v)) > w(t ). The nibble placement can be calculated in time O( V ) for each object. If the function h r and h w are represented by an array including an entry for each node, then this bound corresponds to the input size. We only need to compute the total number of write accesses and, for each node v, the sum of the read and write accesses issued in the subtrees incident on v, i.e., the subtrees into which the tree is partitioned if v is removed from the tree. This can be done by a depth first search algorithm taking time O( E ) = O( V ). Moreover, the placement can be calculated efficiently by the processors of the tree network in a distributed fashion. Here the total number of write accesses and the weight of all subtrees incident on the nodes can be computed in O(height(T )) rounds each of which takes time O(degree(T )). The computation for several objects can be pipelined, which gives time O(( + height(t )) degree(t )) for the placement of all objects in. The most important result of the following theorem is that the nibble strategy minimizes the load on all edges simultaneously, and, hence, also the congestion, regardless of the bandwidths of the edges. Theorem 3.1. The nibble strategy computes a placement that has the following properties. The nibble placement achieves minimum load on each edge. All nodes holding a copy of an object x form a connected subgraph T (x). The load on an arbitrary edge e induced for serving requests to an object x is less or equal than the write contention κ x := P v V hw(v, x) of x. The load on all edges in the connected subgraph T (x) is κ x. 3.2 Step 2: the deletion algorithm removing rarely used copies The algorithm that removes rarely used copies, called the deletion algorithm, is presented in the following. For a node

6 for each edge e do L map( e ) := 0 L acc( e ) := 2 L b ( e ) for l = 0 to height(t ) 1 do for each node v on level l do while (L map( e +) + τ max L acc( e +)) do choose c M(v) arbitrarily M(u) := M(u) {c} M(v) := M(v) \ {c} L map( e +) := L map( e +) + s(c) + κ x(c) δ := L acc( e +) L map( e +) L acc( e +) := L acc( e +) δ L acc( e ) := L acc( e ) δ 9 = ; adjustment 9 = ; movement along e + = (v, u) Figure 5: The upwards phase of the mapping algorithm. possibly reduced ( e denotes the downwards edge incident to v and u). This ensures that in the downwards phase the algorithm does not move too many copies along e in the subtree of v. Note that L acc( e ) may become negative by this adjustment. The details of the upwards phase are described in Figure 5. In the downwards phase consisting of height(t ) rounds all copies are moved along downwards edges. In round l a node v on level l of T moves all his copies in the following way. Suppose v holds a copy c that serves s(c) requests. Then the algorithm searches a free downward child edge e of v, i.e., an edge e with L map( e ) + s(c) + κ x(c) L acc( e ) + τ max, where x(c) denotes the shared data object of which c is a copy. The copy is moved along this edge e and the mapping load of e is increased accordingly. The details of the downwards phase are described in Figure 6. If a free edge can always be found we can make the following observation. Observation 3.3. Suppose a node v has moved all its copies downwards and e is a downward child edge of v. Then either or L map( e ) L acc( e ) + τ max L map( e ) = 0 and L acc( e ) < τ max. 4. THE ANALYSIS First, the following lemma ensures that the mapping algorithm can be executed in the described way. Lemma 4.1. In the downwards phase of the mapping algorithm a free downward child edge can always be found. Proof. To prove the lemma we need the following invariant. Invariant 4.2. Let v denote an arbitrary internal node of the tree network and let E out (E in) denote the set of outgoing (incoming) edges of v. Then the following inequality holds at any time step during the execution of the mapping algorithm, L acc( e ) L map( e ) e E out e E out L acc( e ) L map( e ) + 2 s(c). e E in e E in c M(v) Proof. First, we will prove that the invariant holds at the beginning of the mapping algorithm. Since all mapping loads are initially 0 the invariant simplifies to 2 L b ( e ) 2 L b ( e ) + 2 s(c). e E out e E in c M(v) Recall that initially L acc( e ) = 2 L b ( e ). The inequality holds, since a request issued by a leaf u that is a basic request of an incoming edge is also a basic request of the outgoing edge leading to u. Thus, each request contributing to P e E in L b ( e ) contributes to P e E out L b ( e ), as well. Furthermore, each request served by a copy placed on v at the beginning of the mapping algorithm is basic for some outgoing edge. This means that each request contributing to P c M(v) s(c) also contributes to P e E out L b ( e ). Now, we show that a movement of a copy or an adjustment of loads does not cause the invariant to become invalid. This shows that it holds throughout the mapping algorithm. Consider that a copy c is moved along an outgoing edge e away from node v. Then L map( e ) increases by s(c) + κ x(c). Thus, the left side of the inequality decreases by this value, whereas the right side decreases by 2 s(c) because c is removed from the set M(v). From observation 3.2 follows that s(c) + κ x(c) 2 s(c) holds, and, thus, the invariant holds after the movement. An analogical argument holds if a copy is moved to v along an incoming edge.

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