Greedy Forwarding in Geometric Routing

Updated 14 July 2026

Greedy forwarding is a local routing rule where each node forwards packets to a neighbor that is closer to the destination based on a chosen metric, typically Euclidean distance.
It is applied in geometric and geographic routing to achieve efficient, online and decentralized packet delivery without requiring global topology knowledge.
Practical challenges such as voids and local minima drive the development of recovery strategies like backtracking and link reversal to enhance reliable packet delivery.

Searching arXiv for recent and foundational uses of “Greedy Forwarding” to ground the article in the current literature. Greedy forwarding (GF) is the local routing rule used in geometric and geographic routing in which a node forwards a packet to a neighbor that is closer to the destination than the current node under a chosen metric. In the standard form, when a node $v$ receives a packet for destination $w \neq v$ , it forwards the packet to a neighbor $u$ satisfying $d(u,w) < d(v,w)$ ; the decision is local and uses only the node’s own coordinate, the destination coordinate, and the coordinates of its neighbors (0812.3893). Across the literature, GF is treated as an online, decentralized, memory-less forwarding primitive for Euclidean, hyperbolic, and other metric embeddings, and its main technical tension is between its extreme locality and its susceptibility to voids, dead ends, and local minima (0903.5208).

1. Formal rule and routing model

In geometric routing, GF is defined on an embedded graph $G=(V,E)$ . At a current node $v_i$ , the algorithm selects a neighbor $v_j \in N(v_i)$ that is closer to the destination point $x$ than $v_i$ is. The rule is purely local: it requires the current node, its neighbors, and the destination coordinates, but not global topology knowledge, routing tables, or path computation (0903.5208).

The same condition appears in several equivalent formulations. In Euclidean routing on planar graphs, the forwarding step is written explicitly as

$d(u,w) < d(v,w),$

with the intended metric being the Euclidean $w \neq v$ 0 metric (0812.3893). In the game-theoretic formulation of greedy routing, a path $w \neq v$ 1 from $w \neq v$ 2 to $w \neq v$ 3 is greedy if $w \neq v$ 4, $w \neq v$ 5, each $w \neq v$ 6, and

$w \neq v$ 7

so every hop moves strictly closer to the destination (Lenzner et al., 30 Jan 2026).

A graph is said to support greedy routing if greedy routing delivers each and every packet to the closest member of $w \neq v$ 8 to the packet’s destination; if the destination itself is a node of $w \neq v$ 9, delivery must be to that exact node (0903.5208). This definition separates the forwarding rule from the supporting substrate: GF is only a local decision rule, while guaranteed delivery depends on graph structure, embedding, and recovery mechanisms.

The literature also distinguishes plain GF from related but non-identical local forwarding policies. In MANET and WSN protocol papers, GF usually means forwarding to a neighbor that is closer to the destination than the current node. In some systems, however, the node may first define an eligible candidate set and then apply additional criteria such as trust, residual energy, contention outcome, or delay thresholds before choosing the actual next hop (Mahmood et al., 2018).

2. Geometric substrates and embeddability

The foundational graph-theoretic characterization of greedy-supporting geometric graphs is given in terms of Voronoi cells, vertex regions, and Delaunay edges. For each $u$ 0, the Voronoi cell is

$u$ 1

while for a graph $u$ 2 with neighbor set $u$ 3, the vertex region is

$u$ 4

The exact condition is

$u$ 5

and, equivalently, a geometric graph supports greedy routing if and only if it contains the Delaunay graph as a subgraph (0903.5208). The same work states that the Delaunay graph is the sparsest greedy-supporting graph.

For Euclidean planar graphs, a constructive embeddability result is established for 3-connected planar graphs. Every 3-connected planar graph contains a spanning Christmas cactus subgraph; every Christmas cactus graph can be greedily embedded in $u$ 6; therefore 3-connected planar graphs can be greedily embedded in the Euclidean plane (0812.3893). The embedding places cycles of the Christmas cactus on concentric semicircles centered at the origin and distinguishes up/down moves between semicircles from left/right moves along a semicircle. Routing then splits into the descendant case and the least-common-ancestor-cycle case.

That Euclidean construction is not only existential. It also gives a comparison function

$u$ 7

where $u$ 8 count potential left, right, down, and up edges on the path from source $u$ 9 to target $d(u,w) < d(v,w)$ 0. A move is greedy if it decreases $d(u,w) < d(v,w)$ 1, and the comparison is stated to be consistent with the Euclidean embedding along greedy routes (0812.3893).

A different geometric substrate appears in scale-free networks embedded in hyperbolic metric spaces. There, GF is topology-oblivious: each node knows only its own coordinates, its neighbors’ coordinates, and the destination’s coordinates written on the packet, and forwards to the neighbor closest to the destination in the embedding space (0805.1266). The hidden space is the two-dimensional hyperbolic plane, the connection rule is $d(u,w) < d(v,w)$ 2, and the model yields $d(u,w) < d(v,w)$ 3 with

$d(u,w) < d(v,w)$ 4

This suggests a broad geometric principle: GF performs best when the embedding geometry is congruent with the topology that packets traverse.

3. Failure modes: voids, local minima, and repair

The central failure mode of GF is the local minimum. In MANET terminology, greedy forwarding fails when a node has no neighbor closer to the destination than itself; this condition is called a void or dead end (Mahmood et al., 2018). In GPSR-style descriptions, the same event is the local maximum case, where the forwarding node has no adjacent neighbor that provides strictly better geographic progress (Samundiswary et al., 2010). In structural graph terms, a packet can be trapped at a node that is locally best among its neighbors but not globally closest to the destination (0903.5208).

Several recovery strategies are built specifically to handle this failure. In GPSR and S-GPSR, if greedy forwarding cannot make progress, the protocol falls back to perimeter forwarding. The packet is marked as being in perimeter mode along with the location where greedy forwarding failed, and planar graph traversal with the right-hand rule is used to route counterclockwise around the void until greedy forwarding can resume (Samundiswary et al., 2010).

GRB replaces planarization and face routing with backtracking and local memory. A node sends the packet to its best neighbor; if that neighbor cannot continue, the packet is returned to the sender; the sender marks that neighbor as invalid for that source-destination pair in the Seen Table and tries the next-best unused neighbor; if all candidates are exhausted, the packet backtracks further (Mahmood et al., 2018). The Seen Table records include $d(u,w) < d(v,w)$ 5, $d(u,w) < d(v,w)$ 6, $d(u,w) < d(v,w)$ 7, $d(u,w) < d(v,w)$ 8, and $d(u,w) < d(v,w)$ 9, with $G=(V,E)$ 0 meaning forwarding mode and $G=(V,E)$ 1 meaning the packet has backtracked.

Another family of repair mechanisms is based on link reversal. In neighbor-oblivious and finite-state algorithms for circumventing local minima, a node is stuck if its forwarding set is empty, equivalently if it has no outgoing link in the destination-oriented DAG view. The proposed neighbor-oblivious full and partial link reversal algorithms update only stuck nodes, terminate in finite time on connected networks, yield a destination-oriented DAG, and culminate in one-bit full link reversal and two-bit partial link reversal finite-state variants (Ramachandran et al., 2011).

The same local-minimum pathology reappears in fault-prone Gaussian Interconnected Networks. There, greedy adaptive routing chooses

$G=(V,E)$ 2

with failure when the potential-neighbor set is empty or a hop cap is reached. The paper states that greedy routing can become trapped in local minima where no immediate neighbor appears favorable even though a valid global path exists, and that increasing fault density exponentially increases the likelihood of creating voids that trap the packet in an unrecoverable local minimum (Charrwi et al., 23 Dec 2025).

A large part of the GF literature consists of augmenting the basic distance-decrease rule with additional local decision criteria. In secured GPSR for mobile sensor networks, the usual GPSR choice of the adjacent neighbor with the least distance to the destination is modified by incorporating trust, so that the selected route is the “most trusted distance route” rather than the pure minimum-distance route (Samundiswary et al., 2010). Trust is updated by overhearing the next hop in promiscuous mode within a Trust Update Interval, incrementing or decrementing trust counters according to forwarding behavior and packet-integrity checks.

Energy-aware and multipath variants for WMSNs alter GF more deeply. In AGEM, Smart Greedy Forwarding restricts attention to neighbors inside an adaptive angular sector toward the sink, starting with angle $G=(V,E)$ 3, requiring a minimum of two candidate neighbors, and increasing the angle by $G=(V,E)$ 4 steps until enough candidates are found or $G=(V,E)$ 5 (Medjiah et al., 2012). Within that candidate set, AGEM scores neighbors using

$G=(V,E)$ 6

with the Heinzelman energy model

$G=(V,E)$ 7

Packets from the same source stream are then spread over a ranked best-neighbor set $G=(V,E)$ 8 using stored hop-count history.

GEAMS uses the same twofold policy—Smart Greedy Forwarding and Walking Back Forwarding—but makes the next-hop choice explicitly stream-aware and load-balancing. The paper defines a neighbor objective function

$G=(V,E)$ 9

sorts neighbors into $v_i$ 0, and stores per-source values $v_i$ 1 to map packet hop-count history to neighbor rank (Medjiah et al., 2012). Both AGEM and GEAMS switch to walking-back mode when the number of closer neighbors to the sink is zero.

In duty-cycled sensor networks, the next-hop choice is refined not by security or energy state but by wake-up delay. The one-hop relay selection problem is formulated as

$v_i$ 2

and then relaxed to

$v_i$ 3

The exact optimal policy BF is an MDP policy; the simplified policy SF forwards to the first waking neighbor whose progress exceeds a threshold $v_i$ 4; MF waits for all forwarding neighbors and chooses the one with maximum progress; FF forwards to the first waking neighbor regardless of progress (0907.4870). This formulation makes the delay-progress trade-off explicit.

Contention-based geographic forwarding extends GF into a hybrid relay-election framework rather than a deterministic next-hop rule. In that setting, relays are selected on a hop basis depending on their relative positions, and the node that provides the longest progress towards the destination is preferred, but the actual relay is determined through contention resolution within forwarding regions such as SDR and CDR (Lima et al., 2015). The paper is explicit that this is not plain GF; it is best described as a hybrid contention-based geographic forwarding scheme.

Variant	Additional criterion	Recovery or selection mechanism
GRB	Seen Table validity	Backtracking without planarization
S-GPSR	Trust level	Greedy selection filtered by trust
AGEM	Energy-aware score, adaptive compass	Walking-back forwarding
GEAMS	Energy-aware, stream-aware ranking	Walking Back Forwarding
Duty-cycled SF	Delay-progress threshold $v_i$ 5	Threshold stop/wait rule

5. Coordinate efficiency, control state, and deployment constraints

A major implementation issue in GF is the representation of coordinates or forwarding state. In Euclidean greedy routing for 3-connected planar graphs, the initial coordinate system uses $v_i$ 6 bits per vertex and is then improved to optimal $v_i$ 7 bits by using weight-balanced binary trees, one per heavy path for level information and one per cycle for cycle-position information (0812.3893). The key weights are

$v_i$ 8

and the root-to-leaf code lengths sum telescopically to $v_i$ 9. The stated significance is reduced node storage, packet header size, and communication bandwidth used by routing metadata.

Hyperbolic GF addresses control overhead differently. Because forwarding decisions use only hidden-space geometry rather than explicit topology, the model does not require routing tables, path computation using full network state, or exchange of topology updates to maintain routing correctness (0805.1266). In the growing model, nodes do not change coordinates when topology changes.

At the protocol level, GF often depends on local control state that can become stale. In ETSI GeoNetworking non-area forwarding, GF is sender-based: the sender consults its Location Table, learned from CAMs or Beacons, considers only neighbors closer to the destination than itself, picks the neighbor with the smallest remaining distance, and sends the packet as a unicast frame (Amador et al., 2024). Reliability relies on MAC-layer retransmissions, up to 8 transmissions total by default, but there is no fallback mechanism in the ETSI GF specification if the chosen next hop is unreachable. The paper emphasizes that the Location Table can be stale because Beacons are sent every 3 s unless a CAM was sent in the last second, and LocT entries are retained for up to 20 s without updates. It also reports two mitigations used in Greedy+: invalidating neighbors after 1000 ms and imposing a 1 km range limit on candidate next hops.

Neighbor-oblivious link-reversal repair is another approach to control-state reduction. In that framework, a stuck node need not know exact neighbor states; it only needs a lightweight hello/acknowledgment exchange to determine whether it is stuck, and the dynamic state can be reduced to one bit for full link reversal or two bits for partial link reversal (Ramachandran et al., 2011). This suggests that GF-supporting systems can trade geometric coordinate richness against auxiliary local state, depending on whether the design emphasizes direct greedy reachability or repair after failure.

6. Performance results, strategic models, and terminological scope

Reported performance for GF and GF-based systems is highly sensitive to the supporting geometry and the failure-recovery mechanism. In scale-free networks embedded in hyperbolic metric spaces, the reported static-network performance is extremely high: for $v_j \in N(v_i)$ 0, $v_j \in N(v_i)$ 1, and $v_j \in N(v_i)$ 2, OGF has success ratio $v_j \in N(v_i)$ 3, MGF has $v_j \in N(v_i)$ 4, OGF has maximum stretch 1, and in link-failure experiments MGF achieves $v_j \in N(v_i)$ 5 for $v_j \in N(v_i)$ 6 and $v_j \in N(v_i)$ 7, with average stretch below 1.1 (0805.1266).

In MANET comparisons, GRB is reported to have Packet Delivery Ratio 98.60% versus 97.38% for AODV and 55.52% for DSR, End-to-End Delay 14.98 ms versus 22.17 ms for AODV, and much less routing-control packets than DSR, AODV, and GPSR (Mahmood et al., 2018). Against published GPSR results, GRB’s successful packet delivery rate is reported as 98.98% versus 97.04% for 50 nodes, 1500m x 300m, pause time 0; 99.07% versus 98.16% for 50 nodes, pause time 60; 98.00% versus 97.50% for 112 nodes, pause time 0; and 97.02% versus 95.00% for 200 nodes, pause time 0.

In adversarial WSN settings, S-GPSR is reported to improve delivery ratio substantially over plain GPSR, with delivery ratio near 98% up to 10 malicious nodes for 150-node networks and up to 15 malicious nodes for 200-node networks, improvements of about 25% to 27% over GPSR in more hostile conditions, about 4% higher delay than GPSR in some cases, and about 73% lower routing overhead than GPSR (Samundiswary et al., 2010). By contrast, in ETSI non-area GeoNetworking with DCC, standard GF is reported to fail to reach the Destination Area in both highway and urban scenarios, and in the urban scenario the table gives PDR $v_j \in N(v_i)$ 8 for both CE and NE (Amador et al., 2024). Greedy+ raises CE PDR to 0.3939 and NE PDR to 0.0526, with CE E2E delay 0.0058 s and NE E2E delay 0.0102 s, but the paper concludes that Greedy Forwarding remains poor under congestion and DCC.

A more recent perspective studies GF not as a fixed routing algorithm but as a network-formation constraint. In greedy routing reachability games, agents in a metric space minimize the number of purchased links subject to greedy reachability. For directed edges, equilibria exist, every NE is a social optimum and every social optimum is a NE, and $v_j \in N(v_i)$ 9; for undirected edges in 2D Euclidean space, the price of anarchy is between 1.75 and 1.8, and approximate equilibria outperform the Delaunay triangulation (Lenzner et al., 30 Jan 2026). This suggests that GF can be treated not only as a forwarding rule but also as a structural requirement shaping sparse network design.

The abbreviation “GF” also has distinct meanings outside geometric routing. In dynamic BSTs, GF denotes Greedy Future, a future-aware restructuring rule equivalent to Geometric Greedy, for which the competitive-ratio lower bound is 2 and the additive gap can be $x$ 0 (Sadeh et al., 2023). In online packet forwarding on a directed line, a different greedy algorithm prioritizes packets by

$x$ 1

and for packet lengths 1 or 2 it has exact competitive ratio

$x$ 2

with a general lower bound of $x$ 3 for randomized algorithms (Boyar et al., 6 Mar 2026). These usages are terminologically adjacent but conceptually separate from geographic greedy forwarding.

Taken in its standard networking sense, GF is the minimal local rule “forward to a neighbor closer to the destination.” Its theoretical interest lies in the exact graph and embedding conditions under which such local decisions are globally correct; its practical interest lies in how far one can preserve the simplicity of that rule while adding succinct coordinates, trust filtering, energy balancing, backtracking, walking back, contention resolution, or delay-aware thresholds to cope with the environments in which pure greedy progress alone is insufficient.