A new construction of broadcast graphs

Discrete Applied Mathematics (

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A new construction of broadcast graphs ∗

Hovhannes A. Harutyunyan a , Zhiyuan Li b , a b

Department of Computer Science and Software Engineering, Concordia University, Montreal, QC H3G 1M8, Canada Computer Science and Technology, United International College, Zhuhai, 0086-519000, China

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Article history: Received 31 July 2016 Received in revised form 5 September 2018 Accepted 27 September 2018 Available online xxxx Keywords: Broadcasting Minimum broadcast graph Knodel graph

a b s t r a c t Given a graph G = (V , E) and an originator vertex v , broadcasting is an information disseminating process of transmitting a message from the vertex v to all vertices of the graph G as quickly as possible. A graph G on n vertices is called broadcast graph if the broadcasting from any vertex in the graph can be accomplished in ⌈log n⌉ time. A broadcast graph with the minimum number of edges is called minimum broadcast graph. The number of edges in a minimum broadcast graph on n vertices is denoted by B(n). A long sequence of papers present different broadcast graph constructions and upper bounds on B(n). In this paper, we improve the compounding method and construct new broadcast graphs with a better upper bound on B(n). Consequently, we show that B(n) ≤ (m − k + 1)n − (2m−k+1 − m 2)(m − 2q + 1)2q−1 for n ∈ [2m−1 + 1, 2m − 2 2 +1 ], where n = 2m − 2k − d, m ≥ 5, m−2 k 2 ≤ k ≤ m − 2, 0 ≤ d ≤ 2 − 1, and q = min(⌊ 2 ⌋, k − 2). © 2018 Elsevier B.V. All rights reserved.

1. Introduction Broadcasting is an information disseminating process in an interconnection network originated by one node and spreading a message to all members of the network. Broadcasting is accomplished when every node is informed. The time of the broadcasting process in modern day networks is one of the main measures of its performance. Many studies in the past decades focus on the construction of good network topologies to increase transmitting speed. Many models are developed based on different assumptions of the number of originators, the number of calls which can be made by a node in one time unit, and other characteristics of the network. The classical model of broadcasting makes the following assumptions

• • • •

there is only one originator; each call involves only one informed node and one of its uninformed neighbors; each call requires one time unit; one node can only participate in at most one call per time unit.

A network can be modeled as a connected simple graph G = (V , E), where V is the set of vertices representing the members in the network and E is the set of edges representing the communication links. Definition 1.1. The broadcast scheme of a given graph G from an originator vertex v is a sequence of parallel calls. Each call is represented by an edge with the direction, specifying the sender and the receiver vertices. A broadcast tree is a directed spanning tree of the graph G with the originator at its root and generated by all calls of a broadcast scheme. ∗ Corresponding author. E-mail addresses: [email protected] (H.A. Harutyunyan), [email protected] (Z. Li). https://doi.org/10.1016/j.dam.2018.09.015 0166-218X/© 2018 Elsevier B.V. All rights reserved.

Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Definition 1.2. Given a graph G and an originator vertex v , b(G, v ) defines the minimum time required by any broadcast scheme in G from the originator v . b(G) = max{b(G, v )|v ∈ V (G)} defines the maximum time required broadcasting from any vertex in G. b(G) is called the broadcast time of G. It is easy to see that for any graph G, b(G) ≥ ⌈log n⌉, since the number of informed vertices is at most doubled during each time unit. Note that all logarithms in this paper are of base 2. Definition 1.3. A graph G on n vertices is called a broadcast graph if b(G) = ⌈log n⌉. A broadcast graph with the minimum number of edges is called a minimum broadcast graph (mbg). This minimum number of edges is denoted by the broadcast function B(n). From the application perspective, mbgs represent the cheapest graphs (with the minimum number of edges) where broadcasting can be accomplished in minimum possible time. The study of minimum broadcast graphs and broadcast function B(n) has a long history. Farley, Hedetniemi, Mitchell and Proskurowski have introduced minimum broadcast graphs in [10]. In the same paper, they have defined the broadcast function, determined the values of B(n), for all n ≤ 15 and n = 2k and proved that hypercubes are minimum broadcast graphs. Khachatrian and Harutounian [24,25] and independently Dinneen, Fellows and Faber [7] have shown that Knödel graphs, defined in [26], are minimum broadcast graphs on n = 2k − 2 vertices. Park and Chwa have proved that the recursive circulant graphs on 2k vertices are minimum broadcast graphs [31]. The comparison of these three classes of minimum broadcast graphs can be found in [11]. Besides these three classes, there is no other infinite construction of minimum broadcast graphs. The values of B(n) also have been known for n = 17 [30], n = 18,19 [5,36], n = 20, 21, 22 [29], n = 26 [32,37], n = 27, 28, 29, 58, 61 [32], n = 30,31 [5], n = 63 [28], n = 127 [16], and n = 1023,4095 [33]. Minimum broadcast graphs are difficult to construct due to the gap between the lower and upper bounds on B(n). A long sequence of papers have presented different broadcast graph constructions, and hence, upper bounds on B(n). Upper bounds on B(n) are provided by constructing broadcast graphs with a small number of edges. The authors in [9] have constructed broadcast graphs by combining two or three smaller broadcast graphs and shown that B(n) ≤ 2n ⌈log n⌉ for all n. This construction has been generalized in [6] using up to seven small broadcast graphs. A tight asymptotic bound on L(n)−1 B(n) = Θ (L(n) · n) has been given in [13] by proving that 2 ≤ B(n) ≤ (L(n) + 2)n, where L(n) is the number of consecutive leading 1’s in the binary representation of n − 1. In [24], the compounding method has been introduced which uses vertex cover of graphs. This method has constructed new broadcast graphs by forming the compound of several known broadcast graphs. In [3], the compounding method has been generalized to arbitrary n by using solid vertex cover. A compounding method using center vertices has been introduced in [35] and shown to be equivalent to the method of using solid vertex cover in [8]. The authors in [18] have continued on the line of compounding and introduced a method of also merging vertices. More recently [1], compounding binomial trees with hypercubes has improved the upper bound on B(n) for many values of n. Vertex addition is another approach to construct good broadcast graphs by adding several vertices to existing broadcast graphs. This method has been first studied in [5] for some small values of n. In [16], authors have added one vertex to Knödel graphs on 2k − 2 vertices. The additional vertex has been connected to every vertex in a dominating set of the Knödel graph. In [20], the same method has been applied to generalized Knödel graphs, in order to construct broadcast graphs on any number of vertices. Ad hoc constructions sometimes also provide good upper bounds. This method usually constructs broadcast graphs by adding edges to a binomial tree [13,18,24]. Vertex deletion has been studied in [5]. Several other constructions have been presented in [5,12,13,18,21,34–36]. Lower bounds on B(n) have also been studied in the literature. The authors in [12] have shown B(n) ≥ 2n (⌊log n⌋ − log(1 + ⌈log n⌉ 2 − n)), for any value of n. B(n) ≥ 2n (m − p − 1) has been proved in [27], where m is the length of the binary representation am−1 am−2 ...a1 a0 of n and p is the index of the leftmost 0 bit. This bound has been improved to be B(n) ≥ 2n (m − p − 1 + b), where b = 0 if p = 0 or a0 = a1 = ap−1 = 0 and b = 1 otherwise [19]. The lower bound has been further improved to be B(n) ≥ 2n (m − p + b) in [33]. k2 (2k −1)

Besides the general lower bounds, B(n) ≥ 2(k+1) for n = 2k − 1 has been shown in [28]. The lower bounds on B(2k − 3), B(2k − 4), B(2k − 5) and B(2k − 6) have been given in [32]. The lower bounds on B(2k − 2p ) and B(2k − 2p + 1), where 3 ≤ p < k have been presented in [14]. Better lower bounds for n = 24,25 have been given in [2]. Note that 23 ≤ n ≤ 25 are the only values of n ≤ 32 for which B(n) is unknown. This paper is the extension of [17]. 2. Existing broadcast graphs In 1975, Knödel defined a class of broadcast graphs on even number of vertices [26]. We follow the equivalent definition given in [18,24]. Definition 2.1. Let n be an even number. A Knödel graph is denoted by KGn = (V , E), where the vertex set is V = {v0 , v1 , v2 , . . . , vn−1 } and the edge set is E = {(vx , vy )|x + y ≡ 2s − 1 mod n, 1 ≤ s ≤ ⌊log n⌋}, where 0 ≤ x, y ≤ n − 1. Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Fig. 1. (a) An example of KG14 ; (b) the broadcast scheme from v0 in KG14 .

Fig. 2. A binomial tree BT4 is constructed by connecting the roots r1 and r2 of two binomial trees BT3 . The root of BT4 can be either one of the roots r1 or r2 . The numbers show the broadcast scheme from r2 in BT4 .

By the definition above, if (vx , vy ) ∈ E, we say that vx and vy are connected on the dimension s, and vx is vy ’s neighbor on the dimension s or vice versa. The following broadcast scheme of a Knödel graph on n vertices is called a dimensional broadcast scheme [4]. Every informed vertex calls its neighbor on dimension t at time unit t, for 1 ≤ t ≤ ⌈log n⌉ − 1. Then at the last time unit every informed vertex calls its neighbor on dimension 1. [4] also shows the existence of other dimensional broadcast schemes for Knödel graphs. It is also easy to see that KGn is a ⌊log n⌋ regular graph. Fig. 1 shows an example of Knödel graph on 14 vertices and the dimensional broadcast scheme from vertex v0 . For graph theoretic and communication properties of Knödel graphs, see [11,15,22,23]. Definition 2.2. A binomial tree BTk of degree k is recursively defined for any k ≥ 0. BT0 is a single vertex. The binomial tree BTk consists of two binomial trees BTk−1 having their roots r1 and r2 connected by an edge. Either of r1 or r2 is the root of the binomial tree BTk . Binomial trees are useful for constructing broadcast graphs, since broadcast time of the root in a binomial tree BTk on 2k vertices is ⌈log 2k ⌉ = k. It is easy to see that a binomial tree BTk is a broadcast tree of the broadcast scheme from any vertex in a hypercube Qk . Furthermore, any broadcast tree on n vertices of a broadcast graph on n vertices is a subtree of BT⌈log n⌉ . Fig. 2 presents the binomial tree BT4 , and a minimum time broadcast scheme from the root vertex r2 . The authors in [18] compound the Knödel graph on 2k − 2 vertices with a hypercube to construct a broadcast graph C on n = (2m−k − 2)2k vertices, where 1 ≤ k ≤ m − 3 and m ≥ 4. The construction creates 2k copies of Knödel graph KG2m−k −2 on 2m−k − 2 vertices. Then, it selects the vertices with the same odd label from different copies of KG2m−k −2 to form a hypercube Qk on 2k vertices. So, each vertex vi in the graph C is in a copy of the hypercube Qk if i is odd or not in, otherwise. Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Fig. 3. An example of L, when m = 6 and l = 2. Solid lines and vertices ri form the Knödel graph KG14 . Each binomial tree of degree 2 is replaced by a dotted triangle. A tree vertex w of the binomial tree B0 and the dashed edges show an example of the connections between a non-root vertex and the root vertices. w is connected to the neighbors of the first dimensional neighbor of the root vertex of tree B0 .

3. New construction 3.1. Compounding Knödel graphs with binomial trees In this section, we improve the upper bound on B(n) by introducing a new broadcast graph construction similar to the compounding method in [1] but using Knödel graphs as a base instead of hypercubes. It is clear that any value of n ∈ [2m−1 + 1, 2m − 4] can be represented as n = 2m − 2k − d, where m ≥ 5, 2 ≤ k ≤ m − 2, and 0 ≤ d ≤ 2k − 1. For convenience, we let l = k − 1, n = 2m − 2l+1 − d, 1 ≤ l ≤ m − 3, and 0 ≤ d ≤ 2l+1 − 1 in the following construction. A new broadcast graph L = (V , E) on n = (2m−l − 2)2l vertices, where m ≥ 5 and 0 ≤ l ≤ m − 3, is constructed from m−l 2 − 2 copies of binomial tree of degree l, denoted by B0 , B1 , . . . , B2m−l −3 . The roots of the binomial trees, denoted by ri , form the Knödel graph KG2m−l −2 on 2m−l − 2 vertices, 0 ≤ i ≤ 2m−l − 3. Fig. 3 presents the new construction for m = 6 and l = 2. The next step of the construction is to delete d vertices from L, where 0 ≤ d ≤ 2l+1 − 1, in order to obtain n, the given number of vertices of the broadcast graph. This step can be done by deleting a leaf from any binomial tree repeatedly. Note that we do not delete the root of any binomial tree because it also belongs to KG2m−l−2 . The number of deleted vertices is at most 2l+1 − 1. Then the new construction connects the vertices of binomial trees B0 , B1 , . . . , B2m−l −3 to m − l − 1 vertices of KG2m−l −2 . Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Let ri be the root of the binomial tree Bi and rh be the first dimensional neighbor of ri in KG2m−l −2 . By the definition of Knödel graphs, h ≡ 1 − i mod 2m−l − 2. We connect each non-root vertex w in the binomial tree Bi to all the neighbors of rh in KG2m−l −2 . Let rj denote these neighbors, j + h ≡ j + 1 − i ≡ 2s − 1 mod 2m−l − 2 for all s = 1, 2, . . . , m − l − 1. The edges of E of the graph L are of three types: the edges in the Knödel graph KG2m−l −2 denoted by EH , the edges in all binomial trees B0 , B1 , ..., B2m−l −3 denoted by ET , and the edges between any non-root vertex w ∈ Bi and some vertices in the Knödel graph denoted by EP . By connecting w to some vertices in the Knödel graph, we aim to let w mimic a particular vertex in KG2m−l −2 during the broadcasting. Therefore, the set of edges of the graph L = (V , E) is defined as E = EH ∪ ET ∪ EP , where EP = {(w, rj )|j + 1 − i ≡ 2s − 1 mod 2m−l − 2, 1 ≤ s ≤ m − l − 1, w ∈ Bi \ {ri }, rj ∈ KG2m−l −2 }. From the construction above, the graph L has only these three types of edges. Thus, the number of edges in L is |E | = |EH | + |ET | + |EP |. Knödel graph KG2m−l −2 has

|EH | =

(m − l − 1)(2m−l − 2) 2

edges. All 2m−l − 2 binomial trees B0 , B1 , . . . , B2m−l −3 together have

|ET | = (2m−l − 2)(2l − 1) − d tree edges. To count the number of edges in EP , each binomial tree has 2l − l − 1 vertices except the root and its l neighbors on the first level. In total, the graph L has (2m−l − 2)(2l − l − 1) − d such vertices remaining after removing d leaves. Each of these vertices needs m − l − 1 edges to connect to the vertices in the Knödel graph. And each of the vertices on the first level of any binomial tree (the l neighbors of the root within a binomial tree) needs m − l − 2 additional edges connecting to the vertices of KG2m−l −2 since it is already adjacent to its root. Thus,

|EP | = ((2m−l − 2)(2l − l − 1) − d)(m − l − 1) + (2m−l − 2)l(m − l − 2) Thus, the total number of edges in the graph L is

|E | = (m − l)n − (m + l + 1)2m−l−1 + m + l + 1 In summary, the graph L has |V | = n vertices for any n = 2m − 2l+1 − d, where 1 ≤ l ≤ m − 3 and 0 ≤ d ≤ 2l+1 − 1, 2m−l − 2 vertices and edges of KG2m−l −2 , and every vertex of any binomial tree Bi , 0 ≤ i ≤ 2m−l − 2 is connected to m − l − 1 vertices of KG2m−l −2 . Fig. 3 demonstrates our construction of the graph L for l = 2 and m − l = 4. We first construct a Knödel graph on 24 − 2 vertices. The vertices of KG14 are labeled as r0 , r1 , r2 , . . . , r13 . A binomial tree on 4 vertices is attached to each vertex of KG14 . Then, for example, we connect the vertex w ∈ B0 to the root vertices r0 , r2 and r6 , which are the neighbors of r1 . Theorem 3.1. L is a broadcast graph and for any n = 2m − 2l+1 − d, where m ≥ 5, 1 ≤ l ≤ m − 3, and 0 ≤ d ≤ 2l+1 − 1 B(n) ≤ (m − l)n − (m + l + 1)2m−l−1 + m + l + 1 Proof. It is clear that n ∈ [2m−1 + 1, 2m − 4] for any n above. Thus, ⌈log n⌉ = m. To show that L is a broadcast graph, the broadcast scheme for any originator is described below. 1. If the originator is a root vertex ri in KG2m−l −2 , where 0 ≤ i ≤ 2m−l − 3, then the broadcast scheme of ri consists of the broadcast scheme from ri in KG2m−l −2 concatenated with the broadcast scheme in all binomial tree from their roots. ri first completes broadcasting within the Knödel graph using dimensional broadcast scheme by time unit m − l. So, after time m − l the roots of all binomial trees have the message. Then it takes l time units to broadcast in its binomial tree. Thus, the broadcasting in L completes in m time units. 2. If the originator is a non-root vertex w in Bi , 0 ≤ i ≤ 2m−l − 3 the broadcasting is more complicated. By construction, w is adjacent to all the neighbors of rh , which is the first dimensional neighbor of ri , the root of the binomial tree Bi . Consider the dimensional broadcast scheme described in Section 2 from rh in KG2m−l −2 . rh informs its neighbor on dimension t at time unit t for all t = 1, 2, . . . , m − l. Since w is adjacent to all neighbors of rh , w can play the role of rh in the broadcast scheme from the originator w in L. w informs ith dimensional neighbor of rh at time unit i, for all i = 1, 2, . . . , m − l − 1. Every informed vertex continues broadcasting as in the dimensional broadcast scheme from rh . As a result every vertex in KG2m−l −2 except rh can be informed by the same broadcast scheme from rh in KG2m−l −2 at the same time, which is m − l. Then rh can be informed by a call from ri at the time unit m − l. Note that since the degree of the vertex ri in KG2m−l −2 is m − l − 1, and ri is busy during the first m − l − 1 time units; then ri is idle at the time unit m − l, and so it can call the vertex rh . The first m − l time units of the broadcast scheme from w in L is shown in Fig. 4. Now, every vertex rj , 1 ≤ j ≤ 2m−l − 3 in KG2m−l −2 , which is also the root of Bj , is informed after the time unit m − l. Next, every root rj broadcasts all vertices within its respective binomial tree in the remaining l time units. The broadcasting in L again takes m time units in total. Therefore, L is a broadcast graph. And for any n = 2m − 2l+1 − d ∈ [2m−1 + 1, 2m − 4], where m ≥ 5, 1 ≤ l ≤ m − 3, and 0 ≤ d ≤ 2l+1 − 1 B(n) ≤ (m − l)n − (m + l + 1)2m−l−1 + m + l + 1 Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Fig. 4. The broadcast scheme from w in L in the first m − l time units. ut , 1 ≤ t ≤ m − l − 1 is the t dimensional neighbor of rh . Solid arcs denote the calls of the broadcast scheme from the originator rh in KG2m−l −2 . Dashed arcs denote the calls from the originator w in L. All the other calls of the broadcast scheme from the originator rh in KG2m−l −2 , and the broadcast scheme of the originator w in L are the same.

By substituting l = k − 1, we get the following result. Corollary 3.1. B(n) ≤ (m − k + 1)n − (m + k)2m−k + m + k, where m ≥ 5, 2 ≤ k ≤ m − 2, and 0 ≤ d ≤ 2k − 1 3.2. Combined compounding The binomial tree compounding method described above extends beyond using a Knödel graph on 2l − 2 vertices as a base graph. Finding a better base for compounding is possible. In this section, we combine the binomial tree compounding method with the compounding method [18] and show that this combined compounding method further improves the general upper bound on B(n). The construction of a broadcast graph D on n = (2m−l − 2)2q 2l−q − d vertices, where m ≥ 5, 1 ≤ l ≤ m − 3, 0 ≤ q ≤ l − 1, and 0 ≤ d ≤ 2l+1 , is as follows. First, we construct a broadcast graph C on (2m−l − 2)2q vertices by the hypercube compounding method in [18]. This q construction creates 2q copies of Knödel graph KG2m−l −2 denoted by KG1 , KG2 , . . . , KG2 . Each vertex in the graph C is denoted j by ri indicating the vertex ri with the index i from j’th copy of KGj . The edges of C are of two types: the edges EH in the copies of Knödel graph KG2m−l −2 and the edges EC = {(ris , rit )| j i is odd and (r s , r t ) ∈ Qq }, where Qq is a hypercube of dimension q. The edges in EC connect the vertices ri from different i copies of Knödel graph KG2m−l −2 with the same odd label i and form a hypercube Q of dimension q. Thus, the graph C has j two types of vertices. The vertex ri is in a copy of the hypercube Qq when i is odd, and it is not in, otherwise. The construction, so far, is exactly the same as the hypercube compounding method in [18]. The next step of the combined compounding construction applies the binomial tree compounding method to broadcast j j graph C as a base graph. The construction replaces each vertex ri in C by a binomial tree Bi of degree l − q on 2l−q vertices j with the root ri . As in the binomial tree compounding method, we remove d leaves from the binomial tree(s) to obtain a general value of n. Again, no root vertex is removed since it is a vertex in the graph C . j The construction further adds two types of edges, EP1 and EP2 . Let w be a non-root vertex in the graph D and vertex ri be j w ’s root. Each edge in EP1 connects w to all neighbors of ri ’s first dimensional neighbor if i is odd; each edge in EP2 connects w j to all neighbors of ri , otherwise. Intuitively, every neighbor of each non-root vertex in the base graph C belongs to a distinct copy of the hypercube. Half of the non-root vertices are adjacent to their roots, and the others are adjacent to the neighbors of the first dimensional neighbor of their roots. We count the number of edges |ED | of the graph D separately. The graph has 2q copies of Knödel graph KG2m−l −2 , 2m−l−1 − 1 copies of hypercube Qq and (2m−l − 2)2q copies of binomial tree of degree l − q. Thus,

|EH | =

1 2

(2m−l − 2)(m − l − 1)2q

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|EC | =

1 2

)

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q2q (2m−l−1 − 1)

|ET | = (2l−q − 1)(2m−l − 2)2q To count |EP1 |, every binomial tree has 2l−q − (l − q) − 1 vertices adjacent to m − l − 1 roots in the Knödel graph and l − q vertices on the first level adjacent to m − l − 2 roots (excluding its own root which is already counted in ET ). So,

|EP1 | =

1 2

(m − l − 1)(2l−q − (l − q) − 1)(2m−l − 2)2q +

1 2

(m − l − 2)(l − q)(2m−l − 2)2q

To count |EP2 |, every non-root vertex has m − l − 1 neighbors connected by the edges in EP2 . Therefore,

|EP2 | =

1 2

(m − l − 1)(2l−q − 1)(2m−l − 2)2q

If d leaves are removed, (m − l)d edges are also removed simultaneously because every leaf is associated to m − l edges. Thus, the total number of edges in the graph D is

|ED | = |EH | + |EC | + |ET | + |EP1 | + |EP2 | − (m − l)d = (m − l)n − (m − 2q + 1)(2m−l − 2)2q−1 |ED | is a function of m, l and q, but n = (2m−l − 2)2q 2l−q − d = (2m−l − 2)2l − d, by our representation, is not a function of q. For a particular value of n, there are multiple ways to construct graph D with different values of integer q, 1 ≤ q ≤ l − 1. We analyze the monotonicity of the function |ED | to find when |ED | is the smallest.

⎧ m−1 ⎪ ⎨ |ED | increases when ≤ q ≤ l; 2

⎪ ⎩ |ED | decreases when 0 ≤ q ≤ m − 1 . 2

2 1 3 However, m− is not an integer if m is odd. q has to be either m− or m− . (The two possible 2 2 2 1 m−1 . Therefore, values of q give the same value of |ED |.) Furthermore, q is not necessarily larger than 2 when q = l − 1 < m− 2 2 |ED | is minimized for q = min(⌊ m− ⌋, l − 1). 2 Fig. 5 shows one example of the construction when n = 96. First, n is represented by n = 27 − 25 = (23 − 2)24 , so 2 ⌋, 4 − 1) = 2. Following our construction the value of n has the m − l = 3 and l = 4. The value of q is decided by min(⌊ 7− 2 3 2 2 form (2 − 2)2 2 . Next, the construction creates 4 copies of Knödel graph KG6 denoted by KG1 , KG2 , KG3 , and KG4 . The odd vertices with the same label from different copies of KG6 , for example r11 , r12 , r13 and r14 are selected to form a hypercube Q2 on 4 vertices. Then, a binomial tree B2 on 4 vertices is attached to every vertex in the current graph. Last, we connect the vertex v ∈ B10 and the vertex u ∈ B11 , for example, to the root r51 and r11 , because r51 and r11 are the neighbors of r01 , which is the root of B10 and r11 ’s first dimensional neighbor.

|ED | is minimized for q =

m−2 . 2

Theorem 3.2. B(n) ≤ (m − l)n − (2m−l − 2)(m − 2q + 1)2q−1 , 2 where n = 2m − 2l+1 − d, m ≥ 5, 1 ≤ l ≤ m − 3, 0 ≤ d ≤ 2l+1 − 1, and q = min(⌊ m− ⌋, l − 1). 2

Proof. To show the theorem, we describe a minimum time broadcast scheme for any originator of the graph D. Each vertex can be a root or a non-root, and each binomial tree is in or not in a copy of the hypercube. Therefore, we consider four cases. j

j

1. If the originator is a root vertex ri , and ri is in a compounding hypercube, the broadcast scheme consists of three j individual broadcast schemes. ri informs all vertices inside its own copy of the hypercube in q time units. Then, every copy of the Knödel graph KG2m−l −2 has exactly one informed vertex. This vertex calls all vertices in its copy of KG2m−l −2 in m − l time units. After this step, every copy of the binomial tree has its root informed. The root j informs all other vertices in time unit l − q. The broadcasting from vertex ri in the graph D finishes at time unit q + m − l + l − q = m = ⌈log n⌉. Fig. 6 shows an example of the broadcast scheme originating from vertex r11 . j j j 2. If the originator is a root vertex ri , and ri is not in a compounding hypercube, the originator ri calls all its neighbors j j j j x1 , x2 , . . . , xm−l−1 at time unit m − l − 1. Once each xp in the distinct copy of the hypercube Q p is informed at the time unit p, it immediately starts informing all the vertices in Q p in the next q time units. During this step, every copy of the Knödel graph KGc has the vertex xcp informed at time unit p + q, which is exactly the same time unit as in broadcasting from the vertex ric in the graph D. (If the root ric is the originator in Case 1, it finishes broadcasting in its copy of the hypercube at the time unit q and informs its neighbor xcp in the Knödel graph KGc at the time unit p + q.) Every vertex Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Fig. 5. An example of the graph D when n = 96 = (23 − 2)22 22 . Solid lines are the edges in 4 copies of Knödel graph KG6 . Dashed lines are the edges in 3 copies of hypercube Q2 . Each dotted triangle represents a binomial tree B2 . The dotted dashed lines show the examples of the connections between the non-root vertices and the root vertices. v is connected to the neighbors of r01 , and u is connected to the neighbors of the first dimensional neighbor of its root r11 .

in KGc except ric is informed at the right time unit. Moreover, the vertex ric can be informed by the vertex xc1 at time unit p + q since xc1 is idle at time unit q in the broadcast scheme from the originator ric in KGc . Thus, every root vertex in the base graph C can be informed in the time unit m − l + q. Then, each root vertex informs all vertices in its binomial tree in the next l − q time units. All vertices in the graph D are informed in the time unit m − l + q + l − q = m. See Fig. 7 for the example. j j j 3. If the originator w is a non-root vertex in the binomial tree Bi with root ri , ri is in a copy of the hypercube Q i , and w j j is adjacent to all the neighbors of the vertex rf , which is ri ’s first dimensional neighbor in the Knödel graph KGj . Each j

neighbor of w (also a neighbor of rf ) is in a distinct copy of the hypercube. The originator w can play exactly the same j

role as the vertex rf in broadcasting from the originator in the graph D following Case 2. Fig. 7 presents an example when the vertex r01 is replaced by the vertex u. j 4. If the originator w is a non-root vertex, and w ’s root ri is not in a copy of the hypercube. By the definition of the graph D, j w is adjacent to all the neighbors of the root vertex ri . Each neighbor is again in a distinct copy of the hypercube. Similar j to Case 3, the originator w plays the role of the vertex ri in the broadcast scheme of the graph D. Fig. 7 demonstrates 1 an example when the vertex r0 is replaced by v . Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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Fig. 6. A broadcast scheme for the originator vertex r11 in the graph D on 96 vertices.

Therefore, the graph D is a broadcast graph. For any n = 2m − 2l+1 − d in the interval [2m−1 + 1, 2m − 4], B(n) ≤ (m − l)n − (2m−l − 2)(m − 2q + 1)2q−1 , 2 ⌋, l − 1). where m ≥ 5, 1 ≤ l ≤ m − 3, 0 ≤ d ≤ 2l+1 − 1, and q = min(⌊ m− 2

By substituting l = k − 1 in Theorem 3.2, we get the following corollary. Corollary 3.2. B(n) ≤ (m − k + 1)n − (2m−k+1 − 2)(m − 2q + 1)2q−1 , 2 where n = 2m − 2k − d, m ≥ 5, 2 ≤ k ≤ m − 2, 0 ≤ d ≤ 2k − 1, and q = min(⌊ m− ⌋, k − 2). 2

4. Comparison There are small number of constructions of broadcast graphs for all n ∈ [2m−1 + 1, 2m ]. In particular, upper bound UB1 =

n⌈log n⌉ 2

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Fig. 7. A broadcast scheme originating from the vertex r01 in a graph D on 96 vertices. We can also broadcast from the originator vertex u and v by replacing r01 using the method described in Fig. 4. The vertex u is in a binomial tree attached to a copy of the hypercube, while v is not.

from [9] is the best general bound when n is close to 2m . On the other hand, upper bound UB2 = (m − k + 1)n − 2m−k −

1 2

(m − k)(3m + k − 3) + 2k

from [18] and upper bound k + 1)2m−k + k + 1, 2 where n = 2m − 2k − d, m ≥ 3, 0 ≤ k ≤ m − 3, and 0 ≤ d ≤ 2k − 1

UB3 = (m − k + 1)n − (

m 2

+

from [1] give the best general bound on B(n) when n is close to 2m−1 + 1. Note that [1] presents the upper bound as follows B(n) ≤ (k + 1)n − (t −

k 2

+ 2)2k + t − k + 2,

where 2t < n ≤ (2k − 1)2t +1−k , t ≥ 7 and 2 ≤ k ≤ t + 1 After transforming the above expression following our notation (by substituting t = m − 1 and k = m − k), we get the upper bound UB3 . Note that k in the above bound is not same as k in UB3 . This is clear if one compares the multiple of n in all upper ⌈log n⌉ bound formulas. When m − k + 1 ≤ 2 or m ≤ k ≤ m − 2 then UB2 and UB3 are better bounds than UB1 . 2 Please cite this article in press as: H.A. Harutyunyan, Z. Li, A new construction of broadcast graphs, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.09.015.

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+ 2k )2m−k − 23 m2 + (k + 23 )m + 21 k2 − 21 k − 1 ≥ 0 less edges than UB2 . Note that UB3 always generates ( m 2 Our new broadcast graph constructions are similar to the last two bounds. Thus, we will compare our new bounds with UB3 from [1]. Let our upper bounds given by Corollaries 3.1 and 3.2 are denoted by UB4 and UB5 respectively. We know that ⌈log n⌉ = m. Then, UB1 = m2 n. In order to let UB4 ≤ UB1 and UB5 ≤ UB1 , m − k + 1 ≤ m2 . Therefore, when k ≥ m2 + 1, UB4 ≤ UB1 and UB5 ≤ UB1 . To comparing UB3 with UB4 , k + − 1)2m−k − m + 1 2 2 Since m − k ≥ 2, then UB3 − UB4 > 0. Thus, UB3 > UB4 . Our Corollary 3.1 provides a better upper bound than UB3 . 2 Comparing UB4 with UB5 is more complicated. We know that UB5 also depends on q, and q = min(⌊ m− ⌋, k − 2). From 2 m−2 the previous analysis, UB5 is monotonically decreasing when q ∈ [1, ⌊ 2 ⌋] and is maximized for q = 1. UB3 − UB4 = (

m

UB5 ≤ (m − k + 1)n − (m − 1)(2m−k+1 − 2) Then we compare UB4 and UB5 . UB4 − UB5 ≥ (m − k − 2)(2m−k − 1) m

Since m − k ≥ 2, then UB4 − UB5 ≥ 0. Thus, UB5 is the best known general upper bound when 2m−1 + 1 ≤ n ≤ 2m − 2 2 +1 . Observation Note that in [1] an upper bound on d, the number of deleted vertices from their construction, is not given (Theorem 1.2 of [1]). As a result, one can get more than one different constructions, and in consequence multiple upper bounds on B(n). For example, if d = 2k then Theorem 1.2 of [1] gives two broadcast graph constructions: one compounding an m − k dimensional hypercube and k dimensional binomial trees with 2k vertices deleted; or the other one compounding an m−k−1 dimensional hypercube and k + 1 dimensional binomial trees. The two broadcast graph constructions above give the following upper bounds UBd and UB3 on B(2m − 2k+1 ) respectively. UBd = (m − k)n − (

m 2

−

UB3 = (m − k − 1)n − (

k

2 m 2

+ 2)2m−k − 2m + k + 3 +

k 2

+ 1)2m−k−2 + k + 2

It is clear that the second bound is better than the first one. Our calculations show that assuming an upper bound d ≤ 2k − 1 (d is the number of deleted vertices from their construction) will make Theorem 1.2 (5a) always generating one broadcast graph construction and the upper bound on B(n) is equal to UB3 . 5. Conclusions In the paper we introduce two new broadcast graphs L and D on n vertices for any n ∈ [2m−1 + 1, 2m − 2] and prove a new general upper bound on B(n) ≤ (m − k)n − (m + k + 1)2m−k−1 + m + k + 1, where n = 2m − 2k+1 − d, m ≥ 3, 0 ≤ k ≤ m − 3 and 0 ≤ d ≤ 2k+1 − 1. The comparison shows that the new upper bound is better than the other known m general upper bounds for 2m−1 + 1 ≤ n ≤ 2m − 2 2 +1 . The general upper bound obtained by L is slightly smaller than the recent upper bound from [1] for the same values of n. The improvement is mainly due to the good properties of Knödel graphs, which was used as the base of compounding method. Moreover, changing the base graph to the hypercube-compounding graph given by [18] further improves the upper bound. The combined compounding construction generalizes the binomial tree compounding method and the hypercube compounding method. Graph D is the same as graph L when q = 0 and the same as the hypercube-compounding graph when q = k. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

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