arXiv subjects

Bader Albader

Publications and source records attributed to Bader Albader.

14 recordsLinked to original sources

arXiv2026

Constant-Time Certificate Selection for Local Broadcast Repair in Dense Gaussian and Eisenstein--Jacobi Networks

Dense Gaussian and Eisenstein--Jacobi (EJ) networks are algebraic interconnection networks with compact coordinate balls, fixed degree, and simple modular addressing. A source-centered coordinate-reduction tree gives a non-redundant one-to-all broadcast in the fault-free network, but processor faults can split the tree into multiple healthy components. Unlike search-based repair methods that require a linear scan of the network to select the repair plan, the certificate selectors introduced here operate in $O(1)$ time and $O(1)$ memory, consulting only the fault coordinates. This paper develops this stronger formulation for the one- and two-fault regime: a constant-time certificate selector. Given only the faulty coordinates, the selector classifies the relative fault geometry, chooses a coordinate-reduction orientation, and returns a bounded ordered set of component-crossing repair edges. For dense Gaussian networks $G_k$, every source-free fault set with $|F|\le2$ is repaired with depth at most $k+2$ and with exactly $c-1$ external component-crossing edges for the selected fault-pruned orientation. For dense EJ networks $H_t$, every one-fault placement is repaired within depth $t+1$, and every two-fault placement is repaired within depth $t+2$, again with exactly $c-1$ external repair edges. Exhaustive strict validation confirms the Gaussian selector over $146{,}156$ one- and two-fault cases for $k=5,\ldots,12$ and the EJ selector over $52{,}395$ cases for $t=2,\ldots,8$, with zero failures in connectivity, acyclicity, exact repair count, or depth bound.

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arXiv2026

Re-Rooting-Assisted Edge-Minimum Runtime Repair for Node and Link Failures in Dense Eisenstein--Jacobi Broadcast Networks

One-to-all broadcasting in dense Eisenstein--Jacobi (EJ) networks relies on diameter-level spanning trees that fragment when nodes or links fail. This paper introduces the selected triple $(r,\theta,\Kcomp_{r,\theta})$--a chosen root, a chosen EJ coordinate-reduction orientation, and the healthy component graph induced by that choice--as the fundamental unit of analysis for joint node/link fault recovery. The central result is a necessary and sufficient condition: hybrid repair succeeds if and only if the healthy EJ graph $G'=\Ht-\Fv-\Fe$ is connected. When $G'$ is connected, a spanning tree of $\Kcomp_{r,\theta}$ maps to exactly $c-1$ component-crossing repair edges, which is minimum for the selected pruned tree. Deterministic guarantees include: one/two faulty nodes are always placed on the distance-$t$ boundary by re-rooting; a single failed link is either avoided or repaired by exactly one crossing edge; and the repaired depth satisfies $D_{r,\theta}\le 2t+1$ under shallowest-layer entry selection. A 260,000-trial validation campaign confirms 100\% recovery and substantial repair-edge reduction over fixed-source repair across five network scales up to $N=120601$ nodes, while global-BFS, near-miss, and cap-sensitivity audits clarify the tradeoff between reachability, forwarding-state changes, and ranked root selection.

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arXiv2026

Re-Rooting-Assisted Edge-Minimum Runtime Repair for Node and Link Failures in Dense Gaussian Broadcast Networks

Dense Gaussian networks are degree-four algebraic networks with compact diameter and coordinate-based routing. Their diameter-level broadcast trees are efficient but fragile under node, link, and runtime-discovered faults. This paper develops a runtime recovery framework for dense Gaussian broadcast networks under static node/link faults and mixed faults, plus single-link faults discovered live. The method re-roots the source so known node faults become boundary leaves whenever possible, then filters failed links and repairs gaps by connecting healthy components of the pruned tree. For a selected root with connected healthy component graph, we prove exactly $c-1$ external repair edges are necessary and sufficient. We also prove deterministic single-link repair, give a constant-size boundary-intersection primitive for source selection, derive a link-avoidance exclusion test, and add a local-obstruction bound explaining why high-order cuts vanish as $k$ grows. Experiments over $k\in\{10,25,50,100,200\}$, up to $80{,}401$ nodes, $280{,}000$ static trials, and $15{,}000$ transient trials show 100\% recovery for deterministic and bounded regimes, $99.998\%$ for multi-link faults, and $99.963\%$ for heuristic regimes; non-recovered trials are explained by disconnected components or relocation failure. Re-rooting reduces average repair edges by 80--100\% versus fixed-source repair. Patched Gaussian-link Noxim replays confirm packet-complete execution and show re-rooting reduces repair edges, components, and depth. A completion-cycle audit separates repair benefit from latency: ablations confirm completion time depends on relocation, scheduling, delivery tail, and selector objective, so the paper claims edge-minimum repair rather than universal completion-cycle dominance.

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arXiv2026

Fault-Tolerant Shared-Relay Communication in Circulant Interconnection Networks

Circulant interconnection networks provide symmetric addressing, compact generator descriptions, and uniform local connectivity. This paper maps a degree--redundancy landscape for a fault-tolerant two-hop primitive in directed circulants: given $n$ nodes and degree budget $m$, how large can the worst-case shared-relay multiplicity $R(n,m)$ be? A node is a shared relay for an ordered terminal pair if it has outgoing links to both terminals; an $f$-relay-fault-tolerant circulant requires at least $f+1$ such relays for every pair. The underlying feasibility condition is a cyclic difference-multiplicity condition, which we use as a mathematical tool rather than claim as a new object. The contribution is the network-design framework around this tool: the parameters $R(n,m)$ and $D_f(n)$, a negative theorem for interval circulants, relay-table preprocessing and lookup algorithms, adversarial and random failure guarantees, load-balance scope, certified upper-bound interpretation of heuristic designs, exact small-$n$ calibration, a software lookup-versus-search microbenchmark, and a reproducible study of 526,539 generator sets. The results show that generator choice critically determines worst-case relay survivability: optimized threshold designs achieve $f$-relay-fault tolerance within about $1.16$--$1.63$ of the counting lower bound, while standard interval generators can fail structurally even at much larger degrees.

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arXiv2026

Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Eisenstein--Jacobi Networks

Dense Eisenstein--Jacobi (EJ) networks are degree-six algebraic interconnection networks whose finite quotient geometry is naturally represented by a hexagonal axial-coordinate ball. This paper studies non-redundant one-to-all broadcast repair in the dense EJ network generated by $\alpha=(t+1)+t\omega$, where $t$ is the network diameter. We propose EJ-MOEM, a multi-orientation edge-minimum repair method that evaluates a constant-size family of hexagonal broadcast-tree orientations, selects a fault-aware candidate, contracts the fault-pruned tree into healthy components, and reconnects these components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph: every healthy node receives the message exactly once, no faulty node is used, and the original healthy tree components are preserved. We prove that, for a chosen orientation whose fault-pruned component graph is connected, exactly $c-1$ external repair edges are necessary and sufficient, where $c$ is the number of healthy components. We also prove a depth-certificate theorem for EJ coordinate-reduction trees: every one-fault placement admits a repair of depth at most $t+1$, and every two-fault placement admits a repair of depth at most $t+2$. The proof uses the three-strip representation of EJ hexagons, a sector-suffix attachment lemma, a non-adjacent-sector separation lemma, and a six-direction shielding classification for paired cuts. Extended validation includes exhaustive one- and two-fault enumeration for $t=2,\ldots,12,14,16,18$ (up to $N=1027$ and 525,825 two-fault placements at $t=18$), structured theorem-critical tests through $t=30$, and large random tests through $t=200$, all with 100\% success and no violation of the theorem.

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arXiv2026

Ratio-Independent Three-Cycle Decomposition with Optimal Ordered Local-Switch Cost in Six-Regular Non-Axis Eisenstein--Jacobi Networks

Six-regular simple Eisenstein--Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks. This paper gives a ratio-independent decomposition of every six-regular simple non-axis EJ network into three edge-disjoint Hamiltonian cycles using a canonical ordered local-switch model based on unit-parallelogram exchanges. The admitted $d=1$ branch needs no switches; $d=2$ has optimal total cost four; and for $d=3$ and $d\ge4$ both modified factors attain the component-counting lower bound $d-1$. Factor-local switches commute, so chronological interleaving does not alter the final factors or cost within the model. Orbit normalization identifies the exact domain and excludes the unique normalized non-axis norm-three degeneration. For $d\ge4$, an equal-coordinate alternating lift removes reduced-ratio dependence from the fine diagonal coordinate. A block-chain invariant, exhaustive interior-template lemma, and parity-specific successor permutations certify the unused complement: rank advances by one modulo $4d-6$, and arc and connector bijections prove complete coverage. The certificate uses $O(d)$ seed records and expands to the full edge lists in $O(N)$ time. Deterministic symbolic and full-quotient audits, including a dictionary-free fine-incidence check for every $4\le d\le201$, are provided in the accompanying reproducibility package and are not proof premises.

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arXiv2026

Re-Rooting-Based Fault-Tolerant One-to-All Broadcasting in Dense Eisenstein--Jacobi Networks

Dense Eisenstein--Jacobi networks are degree-six algebraic interconnection topologies with regular structure, vertex symmetry, small diameter, and efficient communication algorithms. These properties make them suitable for parallel and on-chip communication systems in which collective operations such as one-to-all broadcasting are frequent. Existing optimal broadcasting algorithms for dense hexagonal/Eisenstein--Jacobi networks assume fault-free operation. However, a faulty internal forwarding node may interrupt message propagation and prevent complete delivery. This paper proposes a lightweight re-rooting-based fault-tolerant broadcasting method for dense Eisenstein--Jacobi networks. The main idea is to relocate the effective broadcast source to a new source node such that each faulty node is located at graph distance equal to the network diameter from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. We present source-selection algorithms for one- and two-node failures and prove that for any pair of faulty nodes in a dense Eisenstein--Jacobi network there exists a common distance-diameter node that can serve as a valid re-rooted source. The source-selection procedure requires linear time in the network diameter. Equivalently, since $N=3t^2+3t+1$, the selection cost is $O(\sqrt{N})$ in the number of nodes. Since the standard one-to-all broadcast completes in one diameter time and the relocation phase is also bounded by one diameter, the proposed method completes in at most twice the network diameter. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving an explicit counterexample. The proposed approach improves broadcast reliability without constructing redundant spanning trees, backup paths, or additional broadcast structures.

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arXiv2026

Closed-Form and Constant-Time New-Source Selection for Fault-Tolerant Broadcasting in Dense Eisenstein--Jacobi Networks

Fault-tolerant broadcasting in dense Eisenstein--Jacobi networks requires efficient recovery when faulty nodes disrupt the original broadcast structure. A re-rooting-based method guarantees that, for any two faulty nodes, a valid new source exists at maximum graph distance from both faults. However, identifying such a source without scanning the network or testing all boundary candidates remains an open practical problem. This paper presents a closed-form, constant-time algorithm for counting and selecting a valid new source in dense Eisenstein--Jacobi networks under two node faults. The two-fault problem is reduced to a boundary-intersection problem involving the origin and a difference node. The distance-$t$ boundary, where $t$ is the network diameter, is partitioned into six directed sides of the Eisenstein--Jacobi hexagon. Since the network is a quotient structure, intersection equations are solved modulo the defining lattice, requiring evaluation of seven quotient-lattice shifts across all $6\times 6$ side pairs, yielding at most $252$ algebraic systems. The first algorithm counts all valid new sources for faults at $0$ and $A$. The second algorithm selects one valid new source for arbitrary fault pairs by solving translated side-pair systems, verifying each candidate, and shifting back. Each system is either a non-parallel $2\times 2$ linear system with at most one candidate, or a parallel system whose feasible candidates form an integer interval. Both algorithms run in $O(1)$ time under the fixed-word arithmetic model. Computational validation over $500{,}000$ sampled fault pairs and $40{,}000$ re-rooting trials confirms correctness: the selector always returns a valid new source, and the recovered broadcast reaches all non-faulty nodes.

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arXiv2026

Closed-Form and Constant-Time New-Source Selection for Fault-Tolerant Broadcasting in Dense Gaussian Networks

Fault-tolerant broadcasting in dense Gaussian networks is recovered by re-rooting the broadcast at a new source at maximum graph distance from the faulty nodes. This paper extends the re-rooting framework by replacing its boundary-search source-selection step with a quotient-lattice-aware algebraic construction. The first contribution is a constant-time counting method for valid new sources, formulated as an intersection of two diameter-$k$ boundary sets in the Gaussian quotient. The exact count is obtained by a fixed union of side-pair intervals over nine quotient-lattice copies, giving a closed-form procedure without scanning the network or boundary. The second contribution is a shifted direct selector for two arbitrary faulty nodes. Given faulty nodes $A$ and $B$, the problem is translated to $C=\operatorname{mod}_{G_k}(B-A)$, and the selector finds $P$ satisfying $d(P,0)=d(P,C)=k$. For each of nine quotient-lattice shifts, sixteen signed linear systems are checked. Nonparallel systems are solved via Cramer's rule; parallel systems are handled by interval-endpoint selection. At most $9\times16=144$ shifted sign cases are evaluated, giving $O(1)$ selection under the word-RAM model. Validation reports zero count mismatches over $26{,}623$ tested nodes, $500{,}000$ valid outputs over $500{,}000$ sampled fault pairs, and $40{,}000$ successful re-rooted broadcast trials. The shifted selector achieves a $5.92\times$ speedup over boundary search at $k=200$, remaining stable as $k$ increases. These results make new-source selection algebraic, bounded, and independent of network size.

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arXiv2026

Local Fault Repair of Perfect Resource Placements in Dense Gaussian Networks

Perfect resource placement in dense Gaussian networks partitions the network into Lee balls centered at resource nodes. The fault-free placement problem is already classified; this paper studies the complementary post-deployment problem of repairing such placements after resource faults. The paper gives exact local repair theorems for the dense Gaussian placement generated by $t+(t+1)i$; by conjugation and rotation symmetry, the same results hold for the companion generator $(t+1)+ti$. For one failed resource, we prove failure-cell locality, derive the exact replacement number $\rho_G(1)=3$ and $\rho_G(t)=2$ for all $t\ge2$, and prove the sharp minimum-overlap formula $\Omega_G(t)=t+1$ among minimum-size repairs. The overlap lower bound is proved from the corner structure of equal-size Lee balls in the rotated coordinates $u=x+y$ and $v=x-y$, where Gaussian Lee balls become parity-constrained squares. For two failed resources, we prove exact additivity: every pair of failed resource cells requires exactly four local replacements for $t\ge2$, and four always suffice. The two-fault lower bound reduces all relevant resource displacements to two canonical neighboring cases and exhibits four mutually incompatible failed-cell corners in each case. For multi-failure repairs, we prove a general inclusion--exclusion identity for overlap inside the failed region; hence the formula remains exact for arbitrary higher-order dense cores. When a canonical repair instance is certified to have maximum multiplicity three, the identity reduces to the compact correction $\Omega_{\rm extra}=P_2-A-C_3$. A ground-truth audit over 7,494 Gaussian cases recomputes coverage from lattice geometry, verifies all exact formulas, and records reproducible multiplicity witnesses.

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arXiv2026

Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Gaussian Networks

Dense Gaussian networks are degree-four algebraic interconnection networks with compact diameter and simple modular routing. This paper studies non-redundant one-to-all broadcast repair in the dense Gaussian network generated by $\alpha=k+(k+1)i$. We propose multi-orientation edge-minimum repair (MOEM), which evaluates a constant-size family of Gaussian broadcast-tree orientations, selects a fault-aware orientation, contracts the fault-pruned tree into healthy components, and reconnects those components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph, so each healthy node receives the message exactly once and no faulty node is used. We prove that, for a chosen orientation with $c$ fault-pruned components and a connected healthy component graph, the repair step is non-redundant and uses the minimum possible number $c-1$ of external component-repair edges. We also prove that, for every one- or two-fault placement, the MOEM orientation family contains a repair with depth at most $k+2$. The depth proof combines a certificate framework, an explicit four-case off-axis analysis, and a five-component orthogonal-axis certificate. Exhaustive validation for $k=5,\ldots,10$ and large-scale validation through $k=200$ confirm the implementation and show that random two-fault repairs use approximately two external repair edges.

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arXiv2026

Local Fault Repair of Perfect Resource Placements in Eisenstein--Jacobi Networks

Perfect resource placements in dense Eisenstein--Jacobi (EJ) networks partition the network into hexagonal radius-$t$ service cells. This paper studies local repair of such placements after resource failures. For one failed resource, we prove that one replacement cannot cover the failed hexagon and two always suffice, giving $\rho_{\mathrm{EJ}}(t)=2$ for all $t\ge1$. Among minimum-size repairs, the sharp minimum-overlap formula $\Omega_{\mathrm{EJ}}(t)=t^2$ follows from the three-strip geometry of EJ balls. For two failed resources, independent repair gives a four-replacement upper bound, but unlike the Gaussian case EJ repair is not always additive: two infinite neighboring displacement families admit three-replacement repairs, proved optimal by a two-ball impossibility argument. Additive behavior is established algebraically via endpoint-rigidity and diagonal-corridor theorems. For $q$ failed resources, independent canonical repair gives a universal $2q$ upper bound, exact when failed cells are pairwise more than $4t$ apart. Dense cluster subadditivity is proved for infinite four-fault and six-fault families with exact repair numbers four and five, giving savings of four and seven over independent repair. An exact inclusion--exclusion identity governs repeated coverage for arbitrary multi-fault repairs. An audit over 19,400 instances confirms widespread subadditivity. EJ local repair is structurally distinct from the Gaussian case: the one-fault overlap is quadratic, two-fault repair can be non-additive, and clustered repairs reuse replacement balls across multiple failed cells.

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arXiv2026

Re-Rooting-Based Fault-Tolerant Broadcasting in Dense Gaussian Networks

Dense Gaussian networks provide degree-4 interconnection topologies with small diameter and regular structure, making them suitable for efficient one-to-all broadcasting. However, node failures can disrupt the broadcast process when faulty nodes occupy internal forwarding positions. This paper proposes a lightweight fault-tolerant broadcasting method based on dynamic source relocation, or re-rooting. Instead of constructing redundant spanning trees or backup routing structures, the proposed method selects a new source node so that the faulty nodes are located at graph distance $k$, the network diameter, from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. For the single-fault case, the new source is selected directly from the graph-distance-$k$ boundary of the faulty node. For the two-fault case, we prove that for any pair of faulty nodes in $G(k+(k+1)i)$, there exists a node whose graph distance from both faulty nodes is exactly $k$. The source-selection procedure requires $O(k)$ time. Since the original one-to-all broadcast completes in $k$ parallel steps and the relocation distance is at most $k$, the proposed method completes in at most $2k$ steps in the worst case. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving a counterexample in $G(3+4i)$. Simulation results confirm complete delivery to all non-faulty nodes under the tested one- and two-node failure scenarios, while the baseline broadcast may fail when faulty nodes occur at internal forwarding positions.

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arXiv2026

A Unified Constant-Time Switch Rule for Constructing Edge-Disjoint Hamiltonian Cycles in Gaussian Networks

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that when the generator $\alpha=a+bi$ satisfies $\gcd(a,b)=1$, the real and imaginary dimensions directly form two edge-disjoint Hamiltonian cycles. A later construction extended the result to the non-coprime case $\gcd(a,b)=d>1$, but its proof used long node-sequence tables and separate odd/even cases for $d$. This paper gives a unified closed-form construction that covers both $d=1$ and $d>1$, and also covers both odd and even $d$, without separate case tables. In the rectangular representation with $d$ rows and $r=(a^2+b^2)/d$ columns, the construction uses a constant-time local switch rule for each $q=1,2,\ldots,d-1$ at column $a_q=q-1$. Each switch removes two horizontal edges and inserts two vertical edges. The switched horizontal structure forms the first Hamiltonian cycle, while its edge-complement in the Gaussian network forms the second Hamiltonian cycle. Thus, the full edge set is partitioned into two edge-disjoint Hamiltonian cycles. The construction requires $O(d)$ switch-generation time and $O(N)$ time to list the two cycles, where $N=a^2+b^2$. Exhaustive validation for all $1\leq a\leq b\leq 100$, excluding only the degenerate $N=2$ network, and large-scale validation up to $N=3{,}250{,}000$ confirm the construction.

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